JOURNAL OF ALGEBRA ARTICLE NO.
186, 934]969 Ž1996.
Counting Characters in Blocks with Cyclic Defect Groups, I Everett C. DadeU Mathematics Department, Uni¨ ersity of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801 Communicated by Walter Feit Received March 7, 1996
In w2x we introduced a conjecture relating the number of ordinary irreducible characters with a given defect in a given p-block B of a finite group G to the numbers of such characters with the same defect in corresponding blocks b of certain p-local subgroups of G. A similar conjecture for projective blocks of G was studied in w3x. There are a large number of other forms of these conjectures. At least seven different ones are mentioned in w4x! Our goal in this paper is to show that one of those forms, the ‘‘Invariant Projective Conjecture’’ w4, 4.7x, is valid for any projective block B of G having a cyclic defect group D. The exact statement of this form will be found in Conjecture 7.9 below. Its proof for blocks with cyclic defect groups is given in Theorem 7.11 below. When the cyclic defect group D is non-trivial, it has a unique subgroup ˜ of prime order p. After suitable reductions, the invariant projective D conjecture for B follows from Theorem 6.1 below, which states that every irreducible character in B has height zero, and that there is a ‘‘natural’’ bijection G of the set IrrŽ B . of all such characters onto the corresponding ˜ s NG Ž D˜. set IrrŽ B˜. for the unique projective block B˜ of the normalizer G inducing B. The bijection G is ‘‘natural’’ enough to be invariant under conjugation by any element t in any extension group E of G such that t ˜ The construction of G is based on ideas of Feit w6x. fixes both B and B. We use his arguments in Section 4 below to prove Theorem 4.3, which is Theorem 6.1 for ordinary blocks. The transition from ordinary blocks to * This research was supported by Grant DMS 93-02996 from the U.S. National Science Foundation. While writing this paper the author was an associate at the Center for Advanced Study of the University of Illinois. He wishes to thank both the Foundation and the Center for their support of this work. 934 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
CYCLIC DEFECT GROUPS, I
projective blocks is handled using the finite covering groups of w3x. This technique reduces Theorem 6.1 to Theorem 5.3 below, which is a version of Theorem 4.3 for certain covering groups. This last result is derived from Theorem 4.3 using ideas of Laradji w9x. In some sense this proof of Theorem 7.11 is not very pretty. The most elegant demonstration would be based on the observation that the entire theory of blocks with cyclic defect groups is just as valid for projective blocks as it is for ordinary blocks. However, verifying this observation would require a book-length paper redoing that theory for projective blocks. Such an article would be too long for our purposes here. So, just as in w3x, we have contented ourselves with the use of finite covering groups to reduce projective block theory to ordinary block theory. The present paper is only the first of two on this subject. The most difficult form of the conjecture, the ‘‘Inductive Conjecture’’ w4, 5.8x, is also valid for blocks with cyclic defect groups. The proof of this fact depends on some long, delicate calculations with Clifford extensions for both characters and modules. We hope to present it some day in part two of this article. But first we have to write the next paper in the sequence w2x, w3x, giving the background necessary to understand this inductive form of the conjecture. Even the present weaker form of the conjecture is quite useful. The inductive form of the conjecture is equivalent to the invariant projective form when the outer automorphism group OutŽ G . of G is cyclic. This is notably the case if G is a sporadic simple group or a simple alternating group with degree different from 6. So our present theorem proves the strongest form of the conjecture for all those groups G and for all primes p such that the Sylow p-subgroup of G is cyclic. This reduces considerably the work required to verify that conjecture for such G, a verification which is a necessary step in the proposed proof of the conjecture for all groups.
1. TWISTED GROUP ALGEBRAS Our notation for finite groups is quite standard, and follows that of w2x. Let H be any finite multiplicative group. We write I F H to indicate that I is a subgroup of H, and I - H to indicate that the subgroup I is properly contained in H. To say, in addition, that the subgroup I is normal in H, we write I 1 H and I 1 H, respectively. We use exponential notation for conjugation, so that s t s ty1st and I t s ty1 It for any s , t g H and any I F H. The normalizer of any I F H in any J F H is written as N J Ž I ., and the centralizer as C J Ž I .. We denote by w I, J x the subgroup of H generated by all commutators w s , t x s sy1ty1st of elements s g I with elements t g J.
EVERETT C. DADE
We use the same conventions about rings and algebras as in w2x or w3x. So any ring or algebra B is associative, with an identity element 1 B which acts as the identity on any B-module. We write ZŽ B . for the center of B, and UŽ B . for the unit group of B. Our terminology for group-graded algebras and twisted group algebras comes from w3, Sect. 5x, which contains proofs Žor, at least, references to proofs. for any unsupported statement we make about them. Let H be any finite group and let S be any non-zero commutative ring. By an H-graded S-algebra B we mean an S-algebra, also called B, together with an H-grading for B, i.e., a decomposition Bs
of the S-module B as a direct sum of S-submodules whose products in the algebra B satisfy B s B t : B st
for any s , t g H. The identity component B 1 is an S-subalgebra of B containing 1 B . The s-component B s of B is a two-sided B 1-submodule of B for each s g H. If I is any subset of H, then B w I x denotes the two-sided B 1-submodule Bw I x s
of B. When I is a subgroup of H, the submodule B w I x is an S-subalgebra of B containing 1 B . In that case the I-grading Ž1.3. makes B w I x an I-graded S-algebra. For any s g H, the elements of the set B s l UŽ B . are called the graded units of B with degree s . The set GrUŽ B . of all graded units of B is a subgroup of UŽ B .. If B is non-zero, then the map sending any graded unit u of B to its degree degŽ u. is a well-defined homomorphism of the group GrUŽ B . into H with kernel UŽ B 1 .. An H-graded S-algebra is an S-crossed product of H if it is non-zero and contains graded units of every possible degree s g H. From now on we assume that B is a twisted group algebra of H over S, i.e., an S-crossed product of H whose identity component B 1 s S1 B , S
is isomorphic to S as an S-algebra. It follows that each s-component of B is a free S-module Bs s S u , S
CYCLIC DEFECT GROUPS, I
of rank one with any graded unit u g GrUŽ B . of degree s as a basis. Furthermore, the degree epimorphism of GrUŽ B . onto H can be combined with the isomorphism s ¬ s1 B of UŽ S . onto UŽ B 1 . to form an exact sequence deg
GrU Ž B .
XŽ B . : 1 ª UŽ S .
making GrUŽ B . a central extension of UŽ S . by H. This central extension determines the twisted group algebra B to within isomorphisms, and vice versa. If I F H, then the I-graded S-subalgebra B w I x of the twisted group algebra B is a twisted group algebra of I over S. The s-components B w I x s and B s of B w I x and B coincide for any s g I. So do the sets B w I x s l UŽ B w I x. and B s l UŽ B . of graded units with degree s in these two twisted group algebras. It follows that the central extension GrUŽ B w I x. of UŽ S . by I is just the restriction of the central extension GrUŽ B . of UŽ S . by H. There is a well-defined conjugation action of the group H as automorphisms of the S-algebra underlying its twisted group algebra B. In this action any t g H sends any x g B to x t s x u s uy1 xu g B ,
where u is any graded unit of B with degree t . This is related to the conjugation action of t on the group H by the equations t
Ž Bs . s Bs t
B w I x s B w It x ,
which hold for any s , t g H and any I F H. The subalgebra of all elements of B w I x fixed under conjugation by every s g I is just the center Bw I x s ZŽ Bw I x. I
of B w I x for any I F H. We usually consider the group algebra S H of H over S to be a twisted group algebra of H over S, with the H-grading ?
Ž S H . s s Ss
and the s-component
for any s g H. Then the twisted subgroup algebra Ž S H .w I x is just the
EVERETT C. DADE
group algebra S I for any subgroup I F H. Furthermore, any element t g H is also a graded unit of S H with degree t . So the above conjugation action of H on its twisted group algebra S H coincides with the usual action of H on S H induced by conjugation in the group H. Suppose that the ground ring S is a field E. We say that a twisted group algebra B of H over E is totally split Žor that E is a totally splitting field for B ., if E is a splitting field for the E-subalgebra B w I x of B whenever I is a subgroup of H Žsee w3, 6.2x.. In that case E is also a total splitting field for each twisted subgroup algebra B w I x of B. Similarly, we say that that the finite group H is totally split over E, or that E is a total splitting field for H, if E is a splitting field for each subgroup I of H. This happens if and only if the group algebra E H is totally split as a twisted group algebra of H over E.
2. CHARACTERS AND BLOCKS We use the same ground rings as in w3, 7.1x. Ž2.1. Throughout this paper we fix a local principal ideal domain Ž i.e, a real discrete ¨ aluation ring . R. We denote by F the field of fractions of R, by p the unique maximal ideal of R, and by F the residue class field Rrp of R. We assume that F has characteristic zero, and that the characteristic of F is a prime, which we denote by p. Aside from the above objects, w3, 7.1x also fixed a finite group G and a totally split twisted group algebra A of G over F. In the present paper G will be embedded as a normal subgroup in some extension group E, which will play the same role here as G played in w3x. Thus Ž2.2. We fix a finite group E and a totally split twisted group algebra A of E o¨ er the field F. Of course the twisted group algebra A could be the group algebra F E with its natural E-grading Ž1.10., in which case we are talking about a finite group E such that F is a splitting field for each subgroup of E. This is the basic situation in w2, 5.1x, with the group G discussed there replaced by our present E. In general we can apply any result from w3x or w2x to our present situation by replacing the group G in those papers by the present E. For example, if H is any subgroup of E, then A w H x is a split, semi-simple algebra of finite dimension over the field F by w3, 7.2x. We write IrrŽ A w H x. for the finite set of all irreducible F-characters of A w H x. When A is F E, the set IrrŽ A w H x. becomes the set IrrŽ F H . of all irreducible F-characters of H
CYCLIC DEFECT GROUPS, I
discussed in w2x. We denote by 1c the primitive idempotent of ZŽ A w H x. corresponding to a given c g IrrŽ A w H x.. By a theorem of Schur w8, V.24.3x, the degree c Ž1. of c divides the order < H < of H. We write dŽ c . for the defect of c , the largest non-negative integer d such that p d divides < H
c t Ž xt . s c Ž x .
for any x g A w H x. In particular, the normalizer NE Ž H . acts in this way on the set IrrŽ A w H x.. Since any t g H centralizes ZŽ A w H x. under conjugation Žsee Ž1.9.. we see that Ž2.4. If H F E, then conjugation in the twisted group algebra A induces an action of the factor group NE Ž H .rH on the set IrrŽ A w H x.. An H-graded R-suborder T of A w H x is any R-suborder T of the F-algebra A w H x such that Ts
Ý Ž As l T . .
This decomposition is then an H-grading, making T an H-graded R-order whose s-component Ts s As l T is an R-sublattice of A s for any s g H. We know from w3, 7.8x that there is a unique maximal H-graded R-suborder of A w H x containing all such T. When H is E we follow the notation of w3x and call this unique suborder O. Thus Ž2.5.
O is the unique maximal E-graded R-suborder of A.
For arbitrary subgroups H F E, this unique maximal suborder is O w H x by w3, 10.1x. If A is F E, then the suborder O is R E by the remarks after w3, 7.8x. In that case O w H x is R H for any H F E. We know from w3, 7.8x that O is always a twisted group algebra of E over R, and that any graded unit u of O with degree t g E is also a graded unit of A with the same degree t . This and Ž1.7. imply that Ž2.6. The conjugation action of any t g E on the twisted group algebra A of E o¨ er F restricts to the conjugation action of t on the twisted group algebra O of E o¨ er R.
EVERETT C. DADE
So in the future we shall only speak of the ‘‘conjugation action of t g E on O,’’ without specifying whether that conjugation is with respect to A or O. We follow w3, 7.11x in defining the p-blocks of A w H x, for any H F E, to be the blocks of the unique maximal H-graded R-suborder O w H x of A w H x. So the set BlkŽ O w H x. of all blocks of O w H x is also the set of all p-blocks of A w H x. Notice that BlkŽ O w H x. becomes the set BlkŽ R H . of all p-blocks of the group H when A is F E. If b g BlkŽ O w H x., then 1 b denotes the corresponding primitive idempotent of ZŽ O w H x., and v b denotes the central character in b, the unique epimorphism of ZŽ O w H x. onto F as R-algebras sending 1 b to 1 F Žsee w3, 7.9x.. As usual, we say that a character c g IrrŽ A w H x. belongs to a block b g BlkŽ O w H x., or that b contains c , if 1c is a primitive constituent of the idempotent 1 b in ZŽ A w H x., i.e., if 1 b 1c s 1c 1 b s 1c .
This happens if and only if v b is the composition of the epimorphism crc Ž1. of ZŽ O w H x. onto R with the natural epimorphism of R onto its factor ring F s Rrp. It also happens if and only if multiplication by 1 b is the identity map on any simple A w H x-module affording c . We denote by IrrŽ b . the set of all characters c g IrrŽ A w H x. belonging to b. Then IrrŽ A w H x. is the disjoint union of its subsets IrrŽ b . for b g BlkŽ O w H x.. If c g IrrŽ A w H x., then BŽ c . will denote the unique p-block of A w H x containing c . It follows from Ž2.6. and Ž1.8. that conjugation by any t g E restricts to an isomorphism of O w H x onto O w H t x as R-orders. This isomorphism induces a bijection of BlkŽ O w H x. onto BlkŽ O w H t x., sending any p-block b of A w H x to the unique conjugate p-block bt of A w H t x satisfying the equivalent conditions 1 bt s Ž 1 b .
v bt Ž z t . s v b Ž z .
for any z g ZŽ O w H x.. This conjugation restricts to an action of the group NE Ž H . on the set BlkŽ O w H x.. Because the subgroup H 1 NE Ž H . centralizes ZŽ O w H x. by Ž1.9., it fixes every p-block of A w H x. Hence Ž2.9. If H F E, then conjugation in the twisted group algebra A induces an action of the factor group NE Ž H .rH on the set BlkŽ O w H x.. We should remark that t
B Ž c t . s B Ž c . g Blk Ž O w H t x .
for any c g IrrŽ A w H x. and t g E. This follows easily from Ž2.3., Ž2.7., and Ž2.8..
CYCLIC DEFECT GROUPS, I
In view of Ž1.9. the central character v b in any p-block b of A w H x is an epimorphism of O w H x H s ZŽ O w H x. onto the field F of characteristic p. So Green’s theory w7x for the H-order O w H x gives us a unique H-conjugacy class DefŽ b . of p-subgroups of G Žsee w3, 9.3x.. The members of DefŽ b . are called the defect groups of b. They are the minimal elements under inclusion in the family D Ž b . of all subgroups P F H such that the HŽ w x P . of the usual trace map tr PH : O w H x P ª image O w H x H P s tr P O H H O w H x satisfies the equivalent conditions 1b g O w H x P
vbŽ Ow H x P . s F . H
The common order of these defect groups has the form p dŽ b., where the non-negative integer dŽ b . is the defect of b. When A is F E these objects are the usual defect groups and defect of the p-block b of H, as defined in w5x. The height of any character c g IrrŽ A w H x. is the integer h Ž c . s dŽ BŽ c . . y dŽ c . .
This integer is non-negative by w3, 9.10x. It agrees with the usual height of the irreducible F-character c of H, as defined in w5x, when A is F E. We recall the definition of induced blocks from w3, 10.5x. Let K be any Ow H x w x subgroup of H. We denote by pr O w K x the projection of O H onto its w x summand O K in the direct sum decomposition
˙ Ow H y K x. Ow H x s Ow K x q By w3, 10.4x this projection is an identity-preserving R-linear map sending ZŽ O w H x. into ZŽ O w K x.. DEFINITION 2.12. Let K be any subgroup of a subgroup H F E, and let b be any p-block of A w K x. If the composite map Ow H x vb (pr O w K x : ZŽ O w H x . ª F
is an epimorphism of R-algebras, then it is the central character v b in a unique p-block b of A w H x. In that case we say that b is induced by b and write b s b A w H x. Otherwise the induced block b A w H x is not defined. When A is F E this induction of blocks is just the usual Brauer correspondence b ¬ b H as defined in w5x. For any p-subgroup P of H, we denote by BlkŽ O w H x ¬ P . the set of all b g BlkŽ O w H x. having P as a defect group. We know from w3, 10.15x that Brauer’s First Main Theorem holds for p-blocks of A w H x, i.e., that Ž2.13. Induction of blocks is a bijection of BlkŽ O wNH Ž P .x ¬ P . onto BlkŽ O w H x ¬ P . whene¨ er P is a p-subgroup of a subgroup H F E.
EVERETT C. DADE
The inverse image bP g BlkŽ O wNH Ž P .x ¬ P . of any b g BlkŽ O w H x ¬ P . under this inductive bijection is called the P-Brauer correspondent of b. Thus bP is the unique p-block of A wNH Ž P .x having D as a defect group and inducing b. In fact, the Brauer correspondent bP is the only p-block of A wNH Ž P .x inducing b Žsee w3, 10.15x.. 3. WEAKLY CLOSED SUBGROUPS As we mentioned earlier, the group G we care about is a normal subgroup of the finite group E in Ž2.2.. Instead of the factor group ErG we take an arbitrary image F of E under an epimorphism « with kernel G. Thus for the rest of the paper we fix « , F, and G satisfying Ž3.1. of « .
« is an epimorphism of E onto a group F, and G is the kernel
In other words, we fix an exact sequence E
of finite groups. Any subgroup H F G has both a normalizer NG Ž H . in G and a normalizer NE Ž H . in E. It also has a ‘‘normalizer’’ NF Ž H . in F, namely the image « ŽNE Ž H .. of NE Ž H .. So H really has a whole exact sequence of normalizers }
NE Ž H .
NF Ž H . ª 1.
1 ª NG Ž H .
A similar exact sequence relates the centralizers of H in G and E to the ‘‘centralizer’’ C F Ž H . s « ŽC E Ž H .. of H in F. Notice that the normalizer NF Ž H . depends only on the G-conjugacy class of H, i.e., that NF Ž H t . s NF Ž H .
for any H F G and t g G. This holds because NF Ž H t . s NF Ž H . « Žt . and « Žt . s 1. We are also going to fix B and D satisfying Ž3.5.
B is a p-block of A w G x, and D is a defect group of B.
˜ F D is weakly Then D is a p-subgroup of G. Recall that a subgroup D ˜t of D˜ contained closed in D with respect to E if the only E-conjugate D ˜ itself. The example we always have in mind for D˜ is the unique in D is D subgroup V Ž D . of order p in D when D is non-trivial and cyclic. But for ˜ satisfying the moment we just fix an arbitrary D Ž3.6.
˜ is a weakly closed subgroup of D with respect to E. D
CYCLIC DEFECT GROUPS, I
˜ E, ˜ and F˜ to be the normalizers Then we define G, ˜ s NG Ž D˜ . , G
˜. , E˜ s NE Ž D
˜. F˜ s NF Ž D
˜ in G, E, and F, respectively. So we have an exact sequence of D E˜
F˜ ª 1
of groups corresponding to Ž3.2.. Of course, we also want a p-block B˜ of ˜x corresponding to B. Aw G PROPOSITION 3.9. In the abo¨ e situation there is a unique p-block B˜ of ˜x inducing the p-block B of A w G x. The defect group D of B is also a Aw G ˜ Its normalizer NG Ž D . in G is also its normalizer NG˜Ž D . in defect group of B. ˜ Furthermore, the D-Brauer correspondent BD of B is also the D-Brauer G. ˜ correspondent B˜D of B.
˜ is certainly normal in NG Ž D .. Proof. The weakly closed subgroup D ˜ s NG Ž D˜.. It follows that D is a Hence NG Ž D . is a subgroup of G ˜ with NG˜Ž D . s NG Ž D .. The D-Brauer correspondent BD p-subgroup of G of B is now a p-block of A wNG˜Ž D .x having D as a defect group, i.e., a member of BlkŽ O wNG˜Ž D .x ¬ D .. So Ž2.13. implies that BD induces a p-block ˜x having D as a defect group and BD as its D-Brauer corresponB˜ of A w G dent. The transitivity of block induction w3, 10.6x implies that B˜ induces the same p-block B of A w G x as BD does. So we can complete the proof of the ˜x inducing B is equal proposition by showing that any p-block BX of A w G ˜ to B. ˜ of G˜ is Let DX be any defect group of BX . The normal p-subgroup D contained in DX by w3, 9.8x. Since BX induces B, its defect group DX is contained in some defect group of B by w3, 10.12x. Hence there exists some ˜ F DX F Dt . Because D˜ is weakly closed in D, it must t g G such that D ˜t F Dt . Therefore t lies in G˜ s NG Ž D˜.. be equal to its G-conjugate D y1 X ˜ Ž DX .t , and assume that Thus we may replace D by its G-conjugate ˜ F DX F D. D ˜ is weakly closed in D with respect to G, the above inclusions Since D imply that it is weakly closed in DX with respect to G. So DX is a ˜ such that NG˜Ž DX . s NG Ž DX .. Hence the DX-Brauer correp-subgroup of G X spondent BDX of BX is a p-block of A wNG Ž DX .x having DX as a defect group. By Ž2.13. the p-block B of A w G x induced by BX also has DX as a defect group and BDX X as its DX-Brauer correspondent. But D is a defect group of B. This and the above inclusions imply that DX s D and BDX X s BD . ˜x induced by BD , and the proposiTherefore BX is the p-block B˜ of A w G tion is proved.
EVERETT C. DADE
From now on B˜ will denote the unique block in the preceding proposition. In view of Ž2.4. and Ž2.9. the exact sequence Ž3.2. can be used to transfer the conjugation actions of E on the sets IrrŽ A w G x. and BlkŽ O w G x. to actions of F on those two sets, with any r g F acting in the same way as any t g E such that « Žt . s r . We denote by NE Ž B . and NF Ž B . the stabilizers of B g BlkŽ O w G x. in E and F, respectively. Then we have the exact sequence NE Ž B .
NF Ž B . ª 1
of finite groups. Similarly, we have an exact sequence NE Ž x .
NF Ž x . ª 1
of finite groups formed from the stabilizers NE Ž x . and NF Ž x . of any x g IrrŽ A w G x. in E and F, respectively. It follows from Ž2.10. that the subset IrrŽ B . of IrrŽ A w G x. is invariant under both NE Ž B . and NF Ž B ., and that Ž3.12. If x lies in IrrŽ B ., then the exact sequence Ž3.11. is a subsequence of the exact sequence Ž3.10., i.e., NE Ž x . and NF Ž x . are subgroups of NE Ž B . and NF Ž B ., respecti¨ ely. Evidently we can make similar constructions with the exact sequence ˜x in place of Ž3.2. and B, respectively. Thus Ž3.8. and the p-block B˜ of A w G ˜x. and BlkŽ O w G˜x. induce the conjugation actions of E˜ on the sets IrrŽ A w G actions of F˜ on those two sets. We have an exact sequence NE˜ Ž B˜.
NF˜ Ž B˜. ª 1
of finite groups formed from the stabilizers NE˜Ž B˜. and NF˜Ž B˜. of B˜ g ˜x. in F˜ and F, ˜ respectively. We also have an exact sequence BlkŽ O w G NE˜ Ž x ˜.
NF˜ Ž x ˜. ª 1
of finite groups formed from the stabilizers NE˜Ž x ˜ . and NF˜Ž x˜ . of any ˜x. in E˜ and F, ˜ respectively. The subset IrrŽ B˜. of IrrŽ A w G˜x. is x˜ g IrrŽ A w G invariant under both NE˜Ž B˜. and NF˜Ž B˜.. Furthermore, Ž3.15. The exact sequence Ž3.14. is a subsequence of the exact sequence Ž3.13. whene¨ er x ˜ lies in IrrŽ B˜.. The tight connection between B and B˜ forces their stabilizers in F and F˜ to coincide.
CYCLIC DEFECT GROUPS, I
PROPOSITION 3.16. The stabilizer NF Ž B . of B in F is also the stabilizer ˜ NF˜Ž B˜. of B˜ in F.
˜ and G. This and Proof. Any element t g NE˜Ž B˜. normalizes both G O wG x Ž1.8. imply that conjugation by t leaves invariant the projection pr O ˜x of wG ˜x. Since conjugation by t also fixes B, ˜ it follows from this O w G x onto O w G ˜ and Definition 2.12 that it fixes the p-block B of A w G x induced by B. Hence NE˜Ž B˜. F NE Ž B . and NF˜ Ž B˜. s « Ž NE˜ Ž B˜. . F « Ž NE Ž B . . s NF Ž B . . Now let t be any element in NE Ž B .. Conjugation by t must permute among themselves the defect groups of B. Since these defect groups form a single G-conjugacy class DefŽ B ., there is some r g G such that Dt s D r . Because Ž3.10. is exact, the element r lies in NE Ž B .. It follows that s s try1 lies in NE Ž D . l NE Ž B .. ˜ is weakly closed in D with respect to E, its normalizer E˜ in E Since D ˜ and G. contains NE Ž D .. Thus s is an element of E˜ normalizing both G ˜x to A w G x is s-invariant. It This implies that induction of blocks from A w G ˜x inducing the follows that s must fix the unique p-block B˜ of A w G p-block B s B s of A w G x. Hence s lies in NE˜Ž B˜.. Since r lies in the kernel G of « , we conclude that
« Ž t . s « Ž s . « Ž r . s « Ž s . g NF˜ Ž B˜. . But t was an arbitrary element of NE Ž B .. Therefore NF Ž B . s « Ž NE Ž B . . F NF˜ Ž B˜. , and the proof of the proposition is complete. In view of the above proposition the action of NF˜Ž B˜. on IrrŽ B˜. induced by conjugation in A via the exact sequence Ž3.13. is also an action of the equal group NF Ž B .. In this way both IrrŽ B . and IrrŽ B˜. become NF Ž B .-sets. Thus it makes sense to speak of an isomorphism of IrrŽ B˜. onto IrrŽ B . as NF Ž B .-sets. We are going to show in Theorem 6.1 below that such an ˜ is its isomorphism exists when D is a non-trivial cyclic p-group and D Ž . unique subgroup V D of order p. Before doing so, we must first handle the special case where A s F E.
4. THE ORDINARY THEOREM During the next two sections the twisted group algebra A of E over F will be the group algebra F E with its natural E-grading Ž1.10.. For reasons explained in w2, Sect. 4x, our assumption in Ž2.2. that F is a total
EVERETT C. DADE
splitting field for A s F E, i.e., a splitting field for every subgroup H F E, implies that all the results of block theory over complete valuation rings also hold over our present R. So we shall use those results freely even though their statements in standard sources, such as w5x, refer only to complete valuation rings. We are going to use some standard facts about Brauer roots of blocks. Let P be a defect group of a p-block b of a subgroup H F E. The P-Brauer correspondent bP of b is a p-block of NH Ž P .. Because P is a normal p-subgroup of NH Ž P ., a result of Brauer w5, V.3.10x tells us that there is a unique NH Ž P .-conjugacy class of p-blocks b of PC H Ž P . inducing bP . We call any such b a P-Brauer root of b. The stabilizer NH Ž P, b . of b in NH Ž P . is the inertial subgroup for b . We often denote this inertial subgroup by T Ž b ., following the notation in w5x. The inertial index e Ž b . s NG Ž P , b . : PC H Ž P . depends only on the block b, and not on the choices of P and b . A theorem of Brauer w5, V.5.2x tells us that Ž4.1.
The inertial index eŽ b . is always relati¨ ely prime to p.
Our field F is big enough that we can define the Brauer character f V of any finite dimensional right F H-module V. We denote by H pX the set of all pX-elements in H. Let m be the least common multiple of the orders of the elements in H pX . Since F is a splitting field for each subgroup of H, there is some primitive mth root of unity r in F. Clearly r lies in the integrally closed subring R of F, and its image r s r q p is a primitive mth-root of unity in F s Rrp. So the natural epimorphism of R onto F restricts to an isomorphism of the group ² r : of all mth-roots of unity in F onto the group ² r : of all mth-roots of unity in F. The Brauer character f V is a function from H pX to R. It sends any s g HpX to the sum
f V Ž s . s r 1 q ??? qrn g R of the unique elements r 1 , . . . , rn g ² r : whose respective images r 1 , . . . , rn g ² r : are all the eigenvalues Žcounting multiplicities. of the F-linear transformation ¨ ¬ ¨ s of V. Before going any farther, we should remark that the existence of a primitive mth-root of unity r in F implies that F is a splitting field for F H. Since H is an arbitrary subgroup of E, we conclude that Ž4.2.
F is a total splitting field for E.
We write IBrŽ R H . for the finite set of all Brauer characters f U afforded by simple right F H-modules U. Since any such U is determined
CYCLIC DEFECT GROUPS, I
to within F H-isomorphisms by its Brauer character f U , it follows from Ž4.2. that the characters in IBrŽ R H . correspond one-to-one to the isomorphism classes of absolutely irreducible H-modules in characteristic p. If b is a p-block of H, then IBrŽ b . will denote the subset of all f g IBrŽ R H . belonging to b, in the sense that multiplication by the natural image 1 b of 1 b in F H is the identity map on any simple right F H-module U affording f . Given any f g IBrŽ R H ., we denote by BŽ f . the unique block b g BlkŽ R H . to which f belongs. Our object in this section is to prove THEOREM 4.3. Suppose that A s F E, that the defect group D in Ž3.5. is ˜ is its unique subgroup V Ž D . of order p. non-tri¨ ial and cyclic, and that D Ž Then e¨ ery character in either Irr B . or IrrŽ B˜. has height zero. Furthermore, IrrŽ B . is isomorphic to IrrŽ B˜. as an NF Ž B .-set. To do so we shall use many of the results in w5, Chap. VIIx about blocks with cyclic defect groups. In translating those results from the notation of w5x to that used here, notice that our G, B, and D are the objects with the same names in w5x. We set a s dŽ B ., so that < D < s p a as on w5, p. 269x. Our ˜ F D of order p is called Day 1 in w5x. Its normalizer G˜ in G is subgroup D ˜ and Nay 1 in w5x Žsee w5, pp. 269 and 272x.. So our p-block B˜ called both G is the p-block B˜ s Bay1 in w5x Žsee w5, pp. 270 and 272x.. Our ground rings F, R, and F correspond to K, R, and R, respectively, in w5x. It follows from Ž4.2. that F is a total splitting field for G. Hence we may choose the extension field Kˆ of K defined on w5, p. 270x to be K s F. Then the integral closure Rˆ of R in Kˆ is just R s R, and the block Bˆ of ˆ lying over B is just B. Clearly B and Bˆ have the same inertial index RG e s e Ž B . s e Ž Bˆ. s ˆ e, w x which divides p y 1 by 5, VII.1.3 . The set L of indices defined on w5, p. 276x now has cardinality < L < s Ž p a y 1 . re Ž 4.4. by w5, VII.2.11x. It also satisfies the assumption Ž). on that same page. So we may apply all of the results for the case Kˆ s K in w5, Chap. VIIx to our present situation. We know from w5, VII.2.1x that there are exactly e distinct characters f 1 , . . . , f e in IBrŽ B .. By w5, VII.2.12x there are exactly e q < L < s e q Ž p a y 1.re distinct characters in IrrŽ B .. If < L < ) 1, then IrrŽ B . has a distinguished subfamily of < L < distinct characters xl for l g L. The members of this subfamily are called the exceptional characters in IrrŽ B .. The remaining e distinct non-exceptional characters in IrrŽ B . are denoted by x 1 , . . . , x e . In this case we define x 0 to be the reducible F-character of G given by
EVERETT C. DADE
If < L < s 1, then there are no distinguished exceptional characters in IrrŽ B .. In this case we denote the e q 1 distinct characters in IrrŽ B . by x 0 , x 1 , . . . , x e . We treat x 0 as the exceptional character, and write it as xl , where l is the unique element in L. Then Ž4.5. holds trivially. Notice that this choice of exceptional character is completely arbitrary, since the characters in IrrŽ B . could be renamed so that any one of them becomes x0. For any i, j s 1, . . . , e we denote by d i, j the decomposition number of x i with respect to f j . All the exceptional characters xl g IrrŽ B . have the same value on each pX-element of G by w5, VII.2.12x. So they all have the same decomposition number d 0, j with respect to any character f j g IBrŽ B .. As in w5, Sect. VII.6x we form a graph T with e edges f 1 , . . . , f e and e q 1 vertices x 0 , x 1 , . . . , x e , where the vertex x i is incident with the edge f j if and only if d i, j / 0. By w5, VII.6.4x this graph is a tree, called the Brauer tree of B. We known from Proposition 3.9 that the block B˜ has the same cyclic defect group D and the same D-Brauer correspondent BD as B. It follows that it has the same index of inertia eŽ B˜. s eŽ B . as B. Hence B˜ has the same set L of Ž p a y 1.re indices as B Žsee w5, p. 276x.. So the above ˜ In particular, description of the characters in B also applies to those in B. ˜ ˜ ˜ Ž . IBr B contains exactly e distinct characters f 1 , . . . , f e , while IrrŽ B˜. consists of < L < distinct exceptional characters x ˜l, for l g L, and of e distinct nonexceptional characters x ˜1 , . . . , x˜e , with the usual ambiguities when < L < s 1. Furthermore, the decomposition numbers among these ˜ We denote by x˜0 the characters determine the Brauer tree T˜ of B. exceptional vertex
of this tree. Proof of Theorem 4.3. Every character in IrrŽ B . has height zero by w5, V.2.16x. The same holds for every character in IrrŽ B˜.. So our only problem is to show that IrrŽ B . is NF Ž B .-isomorphic to IrrŽ B˜.. In view of Proposition 3.16 it will suffice to find a bijection of IrrŽ B . onto IrrŽ B˜. preserving the conjugation actions of NE˜Ž B˜. on these two sets. We shall do this in a series of lemmas. The first lemma has some independent interest, since it proves the corresponding result for Brauer Characters. LEMMA 4.7. There is a bijection of IBrŽ B . onto IBrŽ B˜. preser¨ ing the conjugation actions of NE˜Ž ˜ b . on these two sets.
CYCLIC DEFECT GROUPS, I
˜ is the normalizer of the unique subgroup D˜ of order p Proof. Since G ˜ has a trivial in D, any conjugate D s of D by an element s g G y G intersection with D. It follows that the Green correspondence w5, III.5.6x ˜ and D sends the isomorphism classes of finite-dimensional non-profor G jective indecomposable FG-modules V belonging to the block B one-toone onto the isomorphism classes of finite-dimensional non-projective ˜ ˜ belonging to the block B˜ Žsee w5, VII.1.4x.. indecomposable FG-modules V ˜ . of the corresponding V ˜ If V is a simple FG-module, then the socle SŽ V ˜ belongs to a uniquely determined isomorphism class of simple FG-modules. So the function f sending the Brauer character f V of V to the ˜ . is a well-defined map from IBrŽ B . to Brauer character fSŽ V˜ . of SŽ V ˜ IBrŽ B .. Because the above Green correspondence preserves conjugation by elements of NE˜Ž B˜., so does the map f. Furthermore, f is an injection by w5, VII.3.11x. Since the two sets IBrŽ B . and IBrŽ B˜. have the same finite order e, this implies that f is a bijection. Thus the lemma holds. The next lemma will allow us to treat exceptional and non-exceptional characters separately. LEMMA 4.8. If < L < ) 1, then the subset xl4 of all exceptional characters in IrrŽ B . is in¨ ariant under conjugation by elements of NE˜Ž B˜.. If < L < s 1, then the exceptional character xl in IrrŽ B . can be chosen to be NE˜Ž B˜.-in¨ ariant. Similarly, the subset x ˜l4 of all exceptional characters in IrrŽ B˜. is ˜ Ž . < < NE˜ B -in¨ ariant when L ) 1, and can be chosen to be NE˜Ž B˜.-in¨ ariant when < L < s 1. Proof. From the definition of the tree T it is clear that two non-exceptional characters x i and x j in IrrŽ B . are different if and only if they have different decomposition numbers d i, h / d j, h with respect to some f h g IBrŽ B .. In a similar way any non-exceptional character x i and any exceptional character xl in IrrŽ B . have different decomposition numbers d i, h / d 0, h with respect to some f h g IBrŽ B .. But all the exceptional characters in IrrŽ B . have the same decomposition number d 0, h with respect to any character f h g IBrŽ B .. Since conjugation by elements of NE˜Ž B˜. must preserve decomposition numbers, this implies the first statement of the lemma Žsee also w6, 3.2x.. The second statement is a consequence of w6, 3.1x and w6, 2.4Žii.x. The remaining statement is proved similarly. In view of the above lemma we may assume from now on that the exceptional characters form NE˜Ž B˜.-invariant subsets of both IrrŽ B . and IrrŽ B˜.. Then these two subsets are NE˜Ž B˜.-isomorphic. LEMMA 4.9. The subset xl4 of all exceptional characters in IrrŽ B . is isomorphic as an NE˜Ž B˜.-set to the subset x ˜l4 of all exceptional characters in IrrŽ B˜..
EVERETT C. DADE
Proof. If < L < s 1, then xl4 consists of just one NE˜Ž B˜.-invariant character x 0 , and xl4 consists of just one NE˜Ž B˜.-invariant character x ˜0 . Clearly the lemma holds in this case. Now assume that < L < ) 1. In this case the relations between the exceptional characters in IrrŽ B˜. and those in IrrŽ B . are explained on w5, pp. 297]298x, where the present x ˜l are called ul. The arguments given there show that we can rename the exceptional characters in IrrŽ B ., if necessary, and find an integer d s "1, so that the virtual character x ˜l y x˜m of G˜ induces the virtual character
Ž x˜l y x˜m .
s d Ž xl y xm .
of G for each l, m g L. Once d is fixed, these equations determine uniquely the bijection f : xl ¬ x ˜l of the exceptional characters in IrrŽ B . onto those in IrrŽ B˜.. Since conjugation by elements of NE˜Ž B˜. preserves ˜ to G, it must leave this bijection induction of virtual characters from G invariant. Hence f is an isomorphism of xl4 onto x ˜l4 as NE˜Ž B˜.-sets, and the lemma is proved. The subsets of non-exceptional characters in both IrrŽ B . and IrrŽ B˜. are also NE˜Ž B˜.-invariant. We use Lemma 4.7 to show that these two subsets are NE˜Ž B˜.-isomorphic. LEMMA 4.10. The subset x i 4 of non-exceptional characters in IrrŽ B . is isomorphic as an NE˜Ž B .-set to the subset x ˜i 4 of non-exceptional characters in IrrŽ B˜.. Proof. Since the set xl4 of all exceptional characters in IrrŽ B . is NE˜Ž B˜.-invariant, the sum x 0 of those characters is fixed by NE˜Ž B˜.. Hence NE˜Ž B˜. acts by conjugation on the set x 0 , x 1 , . . . , x e 4 of vertices of T as well as on the set f 1 , . . . , f e 4 of edges to T. Because incidence in T is defined in terms of the decomposition numbers d i, j , which must be preserved by these actions of NE˜Ž B˜., we conclude that NE˜Ž B˜. acts as automorphisms of the tree T fixing the exceptional vertex x 0 . Given a non-exceptional vertex x i in the tree T, there is a unique path in T starting at x i and ending at the exceptional vertex x 0 . We let f Ž x i . be the unique edge in this path incident with x i . In this way we define a function f sending the set x i 4 of non-exceptional vertices in T to the set IBrŽ B . of edges in T. Because T is a tree, this function is a bijection. Since the action of NE˜Ž B˜. on T fixes x 0 , it leaves f invariant. Hence f is an isomorphism of x i 4 onto IBrŽ B . as NE˜Ž B˜.-sets. Similarly, the set x ˜i 4 of all non-exceptional vertices of T˜ is isomorphic to IBrŽ B˜. as an NE˜Ž B˜.-set. The NE˜Ž B˜.-sets IBrŽ B . and IBrŽ B˜. are isomorphic by Lemma 4.7. Therefore x i 4 is isomorphic to x ˜i 4 as NE˜Ž B˜.-sets, and the lemma is proved.
CYCLIC DEFECT GROUPS, I
The preceding three lemmas imply that IrrŽ B . is isomorphic to IrrŽ B˜. as an NE˜Ž B˜.-set. In view of Proposition 3.16 the final statement in Theorem 4.3 follows from this. So the proof of that theorem is complete.
5. THE COVERING THEOREM We still assume that A s F E. Instead of taking D to be cyclic, we now make the more complicated assumption that Ž5.1. The defect group D in Ž3.5. is a split extension of a central subgroup Z of E by a non-tri¨ ial cyclic factor group DrZ. Notice that Z is a central p-subgroup of G since it is central in E and contained in the p-subgroup D of G. Furthermore, the fact that Z and DrZ are both abelian implies that D , Z = Ž DrZ . is abelian. ˜ we now take the inverse image in D of the unique subgroup For D ˜ is determined by its properties Ž V DrZ . of order p in DrZ. So D
DrZ s V Ž DrZ . .
˜ ˜t of D˜ contained in D must contain Z s Zt as a Any E-conjugate D ˜ : Z x. Because DrZ is cyclic, this forces D˜t to subgroup of index p s w D ˜ Hence Ž3.6. holds, and G, ˜ E, ˜ F, ˜ and B˜ have all the properties equal D. discussed in Section 3. We fix a linear F-character l of Z. If H is any subgroup of E containing Z, then IrrŽ F H ¬ l. denotes the set of all characters x g IrrŽ F H . lying o¨ er l in the usual sense that 1x 1l s 1x . For any p-block b of H, the intersection IrrŽ b . l IrrŽ F H ¬ l. is denoted by IrrŽ b ¬ l.. In particular, the subset IrrŽ B ¬ l. of IrrŽ B . is defined in this way. Since the character l of the central subgroup Z is fixed under conjugation by any t g E, the subset IrrŽ B ¬ l. of IrrŽ B . is NE Ž B .-invariant. Hence it is NF Ž B .-invariant. ˜ s NG Ž D˜.. The central subgroup Z of G is contained in the normalizer G Hence the set IrrŽ B˜ ¬ l. is defined. As above, the centrality of Z implies that this subset of IrrŽ B˜. is invariant under the conjugation actions of both NE˜Ž B˜. and NF˜Ž B˜.. Since this last group is equal to NF Ž B . by Proposition 3.16, this means that both IrrŽ B ¬ l. and IrrŽ B˜ ¬ l. are NF Ž B .-sets. Our goal in this section is to prove ˜ THEOREM 5.3. Suppose that A s F E, that D satisfies Ž5.1., and that D satisfies Ž5.2.. Fix a character l g IrrŽ F Z .. Then any character in either IrrŽ B ¬ l. or IrrŽ B˜ ¬ l. has height zero. Furthermore, IrrŽ B ¬ l. and IrrŽ B˜ ¬ l. are isomorphic NF Ž B .-sets.
EVERETT C. DADE
We shall use w9, Theorem Ax to derive most cases of this theorem from Theorem 4.3. Before doing so, we must first discuss the relations between ˜ and those of GrZ or GrZ. ˜ p-blocks of G or G Let H be any subgroup of E containing Z. Because F is a total splitting field for H, it is also a total splitting field for the factor group HrZ of H. Hence we can apply the usual theory of p-blocks to those of subgroups of HrZ. The natural epimorphism of H onto its factor group HrZ induces an epimorphism hH of R H onto R Ž HrZ . as R-orders. This epimorphism restricts to an identity-preserving homomorphism of ZŽ R H . into ZŽ R Ž HrZ .. as R-orders. The composition of this homomorphism with the central character v b : ZŽ R Ž HrZ .. ¸ F in any p-block b of HrZ is an identity-preserving homomorphism of ZŽ R H . into F as R-algebras. Hence it is the central character
v b s v b (hH : Z Ž R H . ¸ F
in a unique p-block b of H. This relation between b and b is denoted by inclusion b : b in w5x. ŽThis curious notation comes from Feit’s identification on w5, p. 23x of a block bX with the category of all finitely-generated modules belonging to bX , together with his identification of any module V belonging to b with the module in b inflated from V.. Since Laradji uses Feit’s notation in w9x, we shall follow it here by saying that b contains b when Ž5.4. holds. Because Z is a central p-subgroup of H, we know from w5, V.4.5x that Ž5.5.
Containment is a bijection of BlkŽ R Ž HrZ .. onto BlkŽ R H ..
Since the defect groups of any p-block b of H contain the normal p-subgroup Z of H, we also know from w5, V.4.5x that Ž5.6. If b g BlkŽ R H . contains b g BlkŽ R Ž HrZ .., then each defect group P of b contains Z, and the factor group PrZ is a defect group of b. In one direction containment preserves block induction. PROPOSITION 5.7. Suppose that Z F H F K F E. Let b be the p-block of H containing a p-block b of HrZ. If b induces a p-block b K r Z of KrZ, then b induces the p-block bX of K containing b K r Z . Proof. The R-linear projection pr RR HK of R K onto R H is the identity on H and sends any s g K y H to 0. The projection pr RRŽŽ HK rr ZZ .. of R Ž KrZ .
CYCLIC DEFECT GROUPS, I
onto R Ž HrZ . can be described similarly. It follows that the diagram hK
R Ž KrZ .
RŽ K r Z . pr RŽ HrZ.
RK pr R H
R Ž HrZ .
of epimorphisms of R-modules commutes. Hence so does its restriction to the diagram hK
ZŽ R Ž KrZ ..
RŽ K r Z . pr RŽ HrZ.
RK pr R H
ZŽ R Ž HrZ ..
ZŽ R H .
ZŽ R K .
of homomorphisms of R-modules. By Definition 2.12 the central character in the induced block b K r Z is the composition
v b K r Z s v b (pr RRŽŽ HK rr ZZ .. : Z Ž R Ž KrZ . . ¸ F . This and the commutativety of the above diagram imply that
v b K r Z (hK s v b (pr RRŽŽ HK rr ZZ .. (hK s v b (hH (pr RR HK : Z Ž R K . ¸ F . By Ž5.4. these last equations say that
v bX s v b (pr RR HK : Z Ž R K . ¸ F . So b induces bX , and the proposition is proved. We should remark that containment of blocks need not preserve induction in the other direction, i.e., that the block b in the above proposition could induce a p-block b K of K even though b induces no p-block of KrZ. An example of this occurs when E s G s K is an extra-special group of order p 3 and Z s H is its center. The factor group KrZ is now abelian and non-trivial. It follows that the principal Žand only. p-block b of HrZ s 1 does not induce a p-block of KrZ. However, b is contained in the principal p-block b of Z, which induces the principal p-block b K of K. Proof of Theorem 5.3. We are going to apply w9, Theorem Ax with the group E in that theorem equal to our present G. As the central p-subgroup Z of E, the p-block B of E, and the defect group D of B in that theorem we take our present Z, B, and D, respectively. Then the p-block B in that theorem becomes the unique p-block B of GrZ contained in B.
EVERETT C. DADE
It follows from Ž5.1. that the defect group D is both abelian and a split extension of Z. Furthermore, our present l g IrrŽ F Z . will serve as the linear F-character l of Z in w9x. So all the hypotheses of w9, Theorem Ax are satisfied in our case. That theorem now gives us a map C satisfying Ž5.8.
C is a degree-preser¨ ing bijection of IrrŽ B . onto IrrŽ B ¬ l..
The bijection C depends on the choice of a certain extension of l to a linear F-character m of D, and may not be unique. We can also apply w9, Theorem Ax with the group E in that theorem ˜ and the p-block B of E in that theorem equal to equal to our present G ˜ The central p-subgroup Z, its linear character l, and the our present B. defect group D remain as before. The p-block B in that theorem is now ˜ ˜ As above, that theorem the unique p-block B˜ of GrZ contained in B. ˜ gives us a not necessarily unique map C satisfying Ž5.9.
˜ is a degree-preser¨ ing bijection of IrrŽ B˜. onto IrrŽ B˜ ¬ l.. C
Because Z is contained in G, the exact sequence Ž3.2. induces an exact sequence ErZ
1 ª GrZ
of finite groups. We are going to use this sequence in place of Ž3.2. to form a factor situation satisfying all the hypotheses of Theorem 4.3. Since D is a defect group of B, we know from Ž5.6. that DrZ is a defect group of B. Hence Ž3.5. holds in the factor situation with B and DrZ in place of B and D, respectively. Furthermore, Ž3.6. is satisfied in the factor situation ˜ s V Ž DrZ . in place of D. ˜ with DrZ ˜ and E˜ in the factor situation are the normalizers The equivalents of G
˜ . s NG Ž D˜ . rZ s GrZ ˜ NG r Z Ž DrZ
˜ . s NE Ž D˜ . rZ s ErZ. ˜ NEr Z Ž DrZ Hence
˜ . F˜ s « Ž E˜. s « Ž ErZ plays the same role in both the factor situation and our original situation. So the equivalent of the exact sequence Ž3.8. in the factor situation is
F˜ ª 1.
˜ 1 ª GrZ We claim that Ž5.10.
˜ The equi¨ alent B˜ of B˜ in the factor situation is B.
CYCLIC DEFECT GROUPS, I
˜ Indeed, Proposition 3.9 tells us that B˜ is the unique p-block of GrZ inducing the p-block B of GrZ. It follows from Proposition 5.7 that the ˜ containing B˜ induces the p-block B of G containing B. p-block B˜X of G ˜ Hence B˜ is the unique By Proposition 3.9 this forces B˜X to equal B. ˜ contained in B, ˜ and Ž5.10. is proved. p-block B˜ of GrZ Ž . It is clear from 5.4 that the conjugate B t of B contains the conjugate tZ B of B whenever t g E has image t Z in ErZ. Because containment of blocks is a bijection of BlkŽ R Ž GrZ .. onto BlkŽ RG ., this implies that the equivalent of NE Ž B . for the factor situation is NEr Z Ž B . s NE Ž B . rZ. Hence the equivalent of NF Ž B . for the factor situation is NF Ž B . s « NEr Z Ž B . s « Ž NE Ž B . . s NF Ž B . .
So the equivalent of Ž3.10. is NE Ž B . rZ
NF Ž B . ª 1.
1 ª GrZ
Similarly, the equivalent of the exact sequence Ž3.13. for the factor situation is NE˜ Ž B˜. rZ
NF˜ Ž B˜. ª 1.
˜ 1 ª GrZ
Of course, NF˜Ž B˜. is equal to NF Ž B . by Proposition 3.16. So the above exact sequences and conjugation in ErZ can be used to turn both IrrŽ B . and IrrŽ B˜. into NF Ž B .-sets. The defect group DrZ corresponding to D in the factor situation is ˜ ˜ is cyclic and non-trivial by Ž5.1.. Its subgroup DrZ corresponding to D V Ž DrZ .. So all the hypotheses of Theorem 4.3 are satisfied in the factor situation. That theorem now tells us that Ž5.11. Any character in either IrrŽ B . or IrrŽ B˜. has height zero. Furthermore, there is some isomorphism G of IrrŽ B . onto IrrŽ B˜. as NF Ž B .sets. The bijection C in Ž5.8. sends any character x g IrrŽ B . to a character x g IrrŽ B ¬ l. with the same degree x Ž1. s x Ž1.. Since p dŽ x . and p dŽ x . are the p-parts of the integers < G
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But p dŽ B . and p dŽ B . are the orders of the defect groups D and DrZ of B and B, all respectively. Hence p dŽ B . s < Z < p dŽ B . . Combining this with the preceding equation, we obtain p hŽ x . s p dŽ B .ydŽ x . s p dŽ B .ydŽ x . s p hŽ x . . Hence x and x have the same height hŽ x . s hŽ x ., i.e., the bijection C ˜ of of IrrŽ B . onto IrrŽ B ¬ l. is height-preserving. Similarly, the bijection C ˜ ˜ IrrŽ B . onto IrrŽ B ¬ l. is height-preserving. Since any character in either IrrŽ B . or IrrŽ B˜. has height zero by Ž5.11., this implies that any character in either IrrŽ B ¬ l. or IrrŽ B˜ ¬ l. has height zero. ˜ are unique when the inertial We claim that Laradji’s bijections C and C index eŽ B . of B satisfies e Ž B . ) 1.
To see this, we must plunge into the construction of these bijections in w9, Sect. 2x. Following the arguments used there, we choose a D-Brauer root b for B, and denote by T Ž b . the inertial subgroup NG Ž D, b . of b. The product group DC G Ž D . is equal to C D Ž G . since D is abelian. The order eŽ B . ) 1 of the factor group T Ž b .rC G Ž D . s T Ž b .rDC G Ž D . is relatively prime to p by Ž4.1.. So this factor group acts faithfully by conjugation as a non-trivial pX-group of automorphisms of the abelian p-group D. A result of Zassenhaus w8, III.13.4bx now tells us that D is the direct product D s C D Ž T Ž b . . = D, T Ž b . , where the factor w D, T Ž b .x is non-trivial. The central subgroup Z F D of T Ž B . is contained in the other factor C D ŽT Ž b ... Because DrZ , ŽC D ŽT Ž b ..rZ . = w D, T Ž b .x is a cyclic p-group, and hence is indecomposable, this forces Z to equal C D ŽT Ž b ... Therefore D is equal to Z = w D, T Ž b .x, and the bijection C is unique by w9, Corollary 1x. We know from Proposition 3.9 that B and B˜ have the same defect group D and the same D-Brauer correspondent BD s B˜D . It follows that they have the same inertial index eŽ B . s eŽ B˜.. So eŽ B˜. ) 1 by Ž5.12., and ˜ is also the above arguments can be repeated to show that the bijection C ˜ implies that they preserve conjugaunique. The unicity of both C and C ˜ and fixing both B and B. ˜ Any element tion by any t g E normalizing G ˜ are isomort g NE˜Ž B˜. has these properties. Therefore both C and C phisms of NE˜Ž B˜.-sets. In view of Proposition 3.16, this implies that both of them are NF Ž B .-isomorphisms. Combining them with the NF Ž B .-isomor˜ ( G ( Cy1 of IrrŽ B ¬ l. onto phism G in Ž5.11., we obtain an isomorphism C IrrŽ B˜ ¬ l. as NF Ž B .-sets. Thus Theorem 5.3 holds when Ž5.12. does.
CYCLIC DEFECT GROUPS, I
From now on we assume that Ž5.12. is false, i.e., that e Ž B . s 1.
In this case we are going to apply exceptional character theory for a ˜ suitable trivial intersection subset of G. Because DrZ is cyclic, the set S ˜ is of all elements s g GrZ generating cyclic subgroups ² s : containing D a trivial intersection subset of GrZ with normalizer
˜ . s GrZ. ˜ NG r Z Ž S . s NG r Z Ž DrZ ˜ .1 ˜ Furthermore, S is contained in C G r Z Ž DrZ GrZ. It follows that the } inverse image ˜ F ² s :Z 4 S s s g G ¬ D of S is a trivial intersection subset of G contained in its normalizer
˜ . s G. ˜ NG Ž S . s NG Ž D ˜ has an abelian defect group D and trivial inertial The p-block B˜ of G ˜ index eŽ B . s eŽ B . s 1. Hence it is nilpotent by w1, 1.ex.3x. So w1, 1.2Ž3.x tells us that there is exactly one character f˜ g IBrŽ B˜.. The character x 0 ˜pX of pX-elements of G. ˜ defined in w1, 1.2Ž5.x restricts to f˜ on the set G Since D is abelian, this and w1, 1.4x imply that each character x ˜ g IrrŽ B˜. ˜pX . restricts to f˜ on G Now assume that x ˜ belongs to IrrŽ B˜ ¬ l.. Since x˜ lies over the linear character l of the central p-subgroup Z, it satisfies x ˜ Ž st . s lŽ s . x˜ Ž t .
˜ This and the conclusion of the preceding for any s g Z and t g G. ˜pX of G. ˜ paragraph imply that x restricts to l = f˜ on the subset Z = G ˜ ˜ Ž . Suppose that x s / 0 for some s g G. Since D is a defect group for ˜ ˜ . w x the p-block B˜ s B Ž x of G, Brauer’s Second Main Theorem 5, IV.6.1 ˜ ˜ If sp f Z, tells us that the p-part sp of s must lie in Dt for some t g G. then ² sp Z : is a non-trivial subgroup of the cyclic p-group DtrZ. This ˜trZ s DrZ. ˜ implies that ² s Z : contains V Ž DtrZ . s D Hence s Z g S ˜pX whenever and s g S whenever sp f Z. Of course s lies in Z = G sp g Z. Let x ˜1 , . . . , x˜n be the distinct characters in IrrŽ B˜ ¬ l.. The preceding two paragraphs tell us that all the characters x ˜i have the same restriction ˜pX and vanish outside S j Ž Z = G˜pX .. It follows that the l = f˜ to Z = G difference x ˜i y x˜j vanishes outside the trivial intersection subset S for
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any i, j s 1, . . . , n. We know from Ž5.9. that the order n of IrrŽ B˜ ¬ l. is equal to the order of IrrŽ B˜.. Because B˜ is a block with a non-trivial defect group DrZ, the latter order is at least 2. Hence n G 2. The theory of exceptional characters now gives us an integer d s "1 and distinct characters x 1 , . . . , xn g IrrŽ FG . such that the virtual character x ˜i y x˜j of G˜ induces the virtual character
Ž x˜i y x˜j .
s d Ž xi y x j .
of G for each i, j s 1, . . . , n. Each character x i must lie over some linear F-character l i of the central subgroup Z. We must prove that l i s l. Since n G 2, there is some j s 1, . . . , n with j / i. The condition Ž5.14. holds for both x ˜i and x˜j . This and Ž5.15. imply that
l i Ž s . x i Ž t . y l j Ž s . x j Ž t . s x i Ž st . y x j Ž st . s lŽ s . Ž x i Ž t . y x j Ž t . .
Ž 5.16. for any s g Z and t g G. Because x ˜i Ž1. s f˜Ž1. s x˜j Ž1., it follows from Ž5.15. that x i Ž1. s x j Ž1.. So Ž5.16. for t s 1 tells us that
li Ž s . xi Ž 1. s l j Ž s . xi Ž 1. for all s g Z. The degree x i Ž1. of the irreducible character x i is not zero. Hence we must have l i s l j . So Ž5.16. is equivalent to
l i Ž s . Ž xi Ž t . y x j Ž t . . s lŽ s . Ž xi Ž t . y x j Ž t . .
for any s g Z and t g G. Since x i and x j are distinct irreducible characters of G, there is some t g G with x i Žt . / x j Žt .. Equation Ž5.17. for this t and arbitrary s g Z implies that l i s l. Therefore x i lies in IrrŽ FG ¬ l. for each i s 1, . . . , n. As on w5, p. 298x, the fact that S contains spC G Ž sp . pX whenever it contains s can be combined with Brauer’s Second Main Theorem to show that each of the characters x i belongs to the p-block B of G induced by ˜ containing all the x˜i . Hence xi g IrrŽ B ¬ l. for each the p-block B˜ of G i s 1, . . . , n. It follows from Ž5.8., Ž5.11., and Ž5.9. that the two sets IrrŽ B ¬ l. and IrrŽ B˜ ¬ l. have the same order n. We conclude that x 1 , . . . , xn are exactly the distinct elements in IrrŽ B ¬ l.. Once the sign d s "1 is fixed, Eqs. Ž5.15. determine a bijection GX of IrrŽ B ¬ l. onto IrrŽ B˜ ¬ l. sending x i to x ˜i for each i s 1, . . . , n. It follows that this bijection is invariant under conjugation by any element t g E
CYCLIC DEFECT GROUPS, I
˜ and fixing both B and B. ˜ But any element t g NE˜Ž B˜. has normalizing G X these properties. Hence G is an isomorphism of IrrŽ B ¬ l. onto IrrŽ B˜ ¬ l. as NE˜Ž B˜.-sets. By Proposition 3.16 it is also an isomorphism of NF Ž B .-sets. Thus Theorem 5.3 holds in all cases. 6. THE PROJECTIVE THEOREM Now we return to the general situation of Section 3, where A is an arbitrary totally split twisted group algebra of E over F. Our goal in this section is to prove the following projective version of Theorem 4.3. THEOREM 6.1. Suppose that the defect group D in Ž3.5. is non-tri¨ ial and ˜ is its unique subgroup V Ž D . of order p. Then e¨ ery cyclic, and that D character in either IrrŽ B . or IrrŽ B˜. has height zero. Furthermore, IrrŽ B . is isomorphic to IrrŽ B˜. as an NF Ž B .-set. We shall derive this theorem from Theorem 5.3 for a suitable finite co¨ ering group EU of A in the sense of w3, 6.1x, i.e., for a suitable finite subgroup EU of GrUŽ A . such that degŽ EU . s E in the exact sequence GrU Ž A .
E ª 1.
XŽ A . : 1 ª UŽ F . U
Such an E exists by w3, 6.4x. We follow the notation of w3, Sect. 6x for objects associated with EU . Thus we denote by hU the restriction of deg: GrUŽ A . ¸ E to an epimorphism of EU onto E, and by ZU the kernel EU l UŽ A 1 . s EU l UŽ F .1 A of hU . So we have an exact sequence 1 }
1 ª ZU EU Eª1 making EU a central extension of the finite cyclic group ZU by the finite group E. We know from w3, 6.4x that F is a total splitting field for the finite covering group EU . So F contains a primitive nth root of unity r, where n is the exponent expŽ EU . of EU . As in w9, Sect. 3x, the product EU ² r : of EU with the finite central subgroup ² r :1 A of GrUŽ A . is another finite covering group for A with the same exponent n as EU . However, the intersection EU ² r : l UŽ A 1 . is ² r : ZU s ² r :1 A , which has the same exponent n as EU ² r :. Hence we may replace EU by EU ² r : and assume from now on that exp Ž EU . s exp Ž ZU . . Ž 6.2. We write z U for the unique faithful linear F-character of ZU such that p s z U Žp U .1 A for any p U g ZU . The cyclic central subgroup ZU of EU is the direct product ZU s ZUp = ZUpX U
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of its unique Sylow p-subgroup ZUp and its unique Hall pX-subgroup ZUpX . It follows that its faithful linear F-character z U is the direct product
z U s z pU = z pUX
of its restrictions to faithful linear F-characters z pU of ZUp and z pUX of ZUpX . Furthermore, the central idempotent in the p-block B Ž z U . of ZU containing z U is precisely 1 BŽ z U . s 1z pUX .
The inverse image ŽhU .y1 Ž P . of any p-subgroup P of E is a nilpotent subgroup of EU since the kernel ZU of hU is central in EU . The Sylow p-subgroup of this nilpotent group is the unique p-subgroup P U of EU satisfying ZUp F P U
hU Ž P U . s P.
As in w3, 9.5x, we express this relation between P U and P by saying that the p-subgroup P U of EU co¨ ers the p-subgroup P of E. We use the epimorphism hU : EU ¸ E to turn the group algebra F EU into an E-graded F-algebra with the s-component
Ž F EU . s s
s gE h U Ž s U .s s
for any s g E. Then w3, 6.19x tells us that inclusion m : E* ¨ GrUŽ A . extends by F-linearity to an epimorphism mU of F EU onto A as E-graded F-algebras. Let H be any subgroup of E, and let H U be its inverse image U y1 Žh . Ž H . in EU . Then F H U is the H-graded F-subalgebra Ž F EU .w H x of F EU , and mU restricts to an epimorphism of F H U onto A w H x as H-graded F-algebras. We know from w3, 10.1 and 8.1x that Ž6.6. Composition with the epimorphism mU : F H U ¸ A w H x is a degree-preser¨ ing bijection of IrrŽ A w H x. onto the set IrrŽ F H U ¬ z U . of all characters c U g IrrŽ F H U . lying o¨ er the character z U g IrrŽ F ZU .. We express the above relation between a character c g IrrŽ A w H x. and the corresponding character c U s c ( mU g IrrŽ F H U ¬ z U . by saying that c U co¨ ers c . The epimorphism mU : F H U ¸ A w H x sends R H U onto O w H x by w3, 10.1 and 7.8x. It follows that it sends ZŽ R H U . into ZŽ O w H x.. In fact, it
CYCLIC DEFECT GROUPS, I
sends ZŽ R H U . onto ZŽ O w H x. by w3, 8.6x. We known from w3, 10.2x that Ž6.7. The epimorphism mU : ZŽ R H U . ¸ ZŽ O w H x. induces a bijection of BlkŽ O w H x. onto the set BlkŽ R H U ¬ z U . of all p-blocks BU of H U lying o¨ er the p-block B Ž z U . of ZU . This bijection sends any p-block b of A w H x to the unique p-block bU of H U satisfying the equi¨ alent conditions
mU Ž 1 bU . s 1 b
v bU s v b ( mU : Z Ž R H U . ¸ F .
We express the above relation between b and bU by saying that bU co¨ ers b. LEMMA 6.8. If bU g BlkŽ R H U ¬ z U . co¨ ers b g BlkŽ O w H x., then co¨ ering of characters is a height-preser¨ ing bijection of IrrŽ b . onto the set IrrŽ bU ¬ z pU . of all characters in IrrŽ bU . lying o¨ er z pU . Proof. We know from w3, 8.9x that covering of characters is a bijection of IrrŽ b . onto the set IrrŽ bU ¬ z U . of all characters in IrrŽ bU . lying over z U . This bijection preserves character heights by w3, 9.10x. Hence the only problem is to show that Irr Ž bU ¬ z U . s Irr Ž bU ¬ z pU . .
Suppose that c U g IrrŽ bU .. Then 1 bU 1c U s 1c U . Because bU g BlkŽ O w H U x ¬ z U . lies over BŽ z U ., we have 1 BŽ z U .1 bU s 1 bU . These equations and Ž6.4. imply that 1z pUX 1c U s 1 BŽ z U .1 bU 1c U s 1c U . Thus c U lies over z pUX . It follows that c U lies over z U s z pU = z pUX if and only if it lies over z pU . This completes the proof of both Ž6.9. and the lemma. Any element t U in the finite covering group EU is a graded unit of A with degree t s hU Žt U .. So conjugation Ž1.7. by t has the same effect on A as conjugation by t U g UŽ A .. The epimorphism mU : F EU ¸ A is the identity on EU by definition. Hence it must carry conjugation by t U into conjugation by t , in the sense that U
mU Ž x t . s mU Ž x . U
for any x g F E .
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Evidently the conjugate Ž H U .t of the inverse image H U s ŽhU .y1 Ž H . is the inverse image ŽhU .y1 Ž H t .. It follows from this, Ž6.6., and Ž6.10. that Ž6.11. If c U g IrrŽ F H U ¬ z U . co¨ ers c g IrrŽ A w H x., and t U g EU U U has image t s hU Žt U . g E, then Ž c U .t g IrrŽ F Ž H U .t ¬ z U . co¨ ers c t g t IrrŽ A w H x.. Furthermore, Ž6.7. and Ž6.10. imply Ž6.12. If bU g BlkŽ R H U ¬ z U . co¨ ers b g BlkŽ O w H x., and t U g EU U U has image t s hU Žt U . g E, then Ž bU .t g BlkŽ R Ž H U .t ¬ z U . co¨ ers bt g BlkŽ O w H t x.. Proof of Theorem 6.1. We denote by GU the inverse image ŽhU .y1 Ž G . of G in EU . Evidently the composition « (hU is an epimorphism « U of EU onto F with kernel GU . So we have an exact sequence EU
1 ª GU
of finite groups. We are going to verify the hypotheses of Theorem 5.3 in the co¨ ering situation where this exact sequence replaces Ž3.2.. For the equivalent of the p-block B in the covering situation we take the p-block BU of GU covering the present p-block B of A w G x. We know from w3, 9.6x that the p-subgroup DU of EU covering the defect group D of B is a defect group of BU . So we can choose DU as the equivalent of D in Ž3.5. for the covering situation. For the central p-subgroup Z of E in Ž5.1. for the covering situation we take the central p-subgroup ZUp of EU . Since ZUp is central in DU , and DU rZUp , D is cyclic, DU is abelian. Our condition Ž6.2. implies that the p-subgroup DU of EU containing ZUp has the same exponent as the Sylow p-subgroup ZUp of ZU . Because DU is abelian, it follows that DU is a split extension of ZUp . Thus Ž5.1. holds in the covering situation. ˜ in the covering situation we take the inverse image D˜U For the group D ˜ s V Ž D . under the epimorphism hU : DU ¸ D. Then Ž5.2. holds in of D ˜U contains the kernel ZUp of the above the covering situation. Since D ˜ It follows epimorphism, it is the unique p-subgroup of EU covering D. ˜ ˜ Žsee w3, 2.5x. that the equivalents of E and G in the covering situation are the inverse images
˜U . s Ž hU . E˜U s NEU Ž D
˜U s NGU Ž D˜U . s Ž hU . G
Ž G˜. ,
˜. and G˜ s NG Ž D˜. in our original situation. respectively, of E˜ s NE Ž D Hence the equivalent of the exact sequence Ž3.8. in the covering situation
CYCLIC DEFECT GROUPS, I
is the exact sequence «U
F˜ ª 1
1 ª E˜U
of finite groups. By Proposition 3.9 the equivalent of B˜ in the covering situation is the ˜U inducing BU . unique p-block B˜U of G
˜U ¬ z U . and co¨ ers B˜ g LEMMA 6.14. The block B˜U lies in BlkŽ RG ˜x.. BlkŽ O w G ˜U ¬ z U . covering B. ˜ Since Proof. Let B˜X be the unique block in BlkŽ RG X U ˜ ˜ w x B induces B, its cover B induces the cover B of B by 3, 10.10 . Hence ˜U inducing BU . That is the lemma. B˜X is the unique p-block B˜U of G Since the p-block BU of GU covers the p-block B of A w G x, and covering of blocks is a bijection of BlkŽ O w G x. onto BlkŽ RGU ¬ z U ., it follows from Ž6.12. that an element t U g EU fixes BU under conjugation if and only if its image t s hU Žt U . in E fixes B under conjugation. Hence the equivalent of NE Ž B . in the covering situation is the inverse image N E U Ž BU . s Ž h U .
Ž NE Ž B . . .
The equivalent of NF Ž B . in that situation is the group
« U Ž NEU Ž BU . . s « Ž NE Ž B . . s NF Ž B . . So the equivalent of the exact sequence Ž3.10. is the exact sequence NEU Ž BU .
NF Ž B . ª 1.
1 ª GU
As usual, we can use this sequence to transfer the conjugation action of NEU Ž BU . on IrrŽ BU . to a conjugation action of NF Ž B . on that set. Because conjugation by any element of EU fixes the character z pU of the central subgroup ZUp , the subset IrrŽ BU ¬ z pU . of IrrŽ BU . is invariant under this action of NF Ž B .. Thus IrrŽ BU ¬ z pU . becomes an NF Ž B .-set. Clearly Ž6.11. implies that Ž6.15. The co¨ ering bijection of IrrŽ B . onto IrrŽ BU ¬ z pU . in Lemma 6.8 for H s G preser¨ es the actions of NF Ž B . on those two sets.
˜U covering the Applying the above arguments with the p-block B˜U of G U ˜ ˜ p-block B of A w G x in place of the p-block B* of G covering the p-block B of A w G x, we see that the equivalent of NE˜Ž B˜. for the covering situation is the inverse image NE˜U Ž B˜U . s Ž hU .
Ž NE˜ Ž B˜. .
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and that the equivalent of the exact sequence Ž3.13. is NE˜U Ž B˜U .
NF˜ Ž B˜. ª 1.
We know from Proposition 3.16 that the group NF˜Ž B˜. is equal to NF Ž B .. So the above exact sequence can be used to transfer the conjugation action of NE˜U Ž B˜U . on IrrŽ B˜U . to one of NF Ž B . on that set. Both these actions leave invariant the subset IrrŽ B˜U ¬ z pU . of IrrŽ B˜U .. Thus IrrŽ B˜U ¬ z pU . becomes an NF Ž B .-set. It follows from Ž6.11. that Ž6.16. The co¨ ering bijection of IrrŽ B˜. onto IrrŽ B˜U ¬ z pU . in Lemma ˜ preser¨ es the actions of NF Ž B . on these two sets. 6.8 for H s G For the linear F-character l of Z in the covering situation we take the linear F-character z pU of ZUp . Then all the hypotheses of Theorem 5.3 for the covering situation are satisfied. That theorem tells us that Ž6.17. Any character in either IrrŽ BU ¬ z pU . or IrrŽ BU ¬ z pU . has height zero. Furthermore, there is an isomorphism GU of IrrŽ BU ¬ z pU . onto IrrŽ B˜U ¬ z pU . as NF Ž B .-sets. Now we can finish the proof of Theorem 6.1. By Lemma 6.8 any character x g IrrŽ B . has the same height as its cover x U g IrrŽ BU ¬ z pU .. The latter character has height zero by Ž6.17.. Hence so does x . Similarly, any character x ˜ g IrrŽ B˜. has height zero because its cover x˜U g IrrŽ B˜U ¬ U z p . has height zero. We know from Ž6.15. that covering of characters is an isomorphism of IrrŽ B . onto IrrŽ BU ¬ z pU . as NF Ž B .-sets. From Ž6.17. we obtain an isomorphism GU of IrrŽ BU ¬ z pU . onto IrrŽ B˜U ¬ z pU . as NF Ž B .-sets. Finally, Ž6.16. tells us that the inverse of the covering bijection is an isomorphism of IrrŽ B˜U ¬ z pU . onto IrrŽ B˜. as NF Ž B .-sets. Hence IrrŽ B . and IrrŽ B˜. are isomorphic NF Ž B .-sets, and Theorem 6.1 is proved. 7. THE INVARIANT PROJECTIVE CONJECTURE For blocks with cyclic defect groups the invariant projective form of the conjectures in w2]4x is an easy consequence of Theorem 6.1. To see this we first have to describe that conjecture. We follow the notation of w3, Sect. 1x for p-chains C of G, i.e., for strictly increasing chains C : P0 - P1 - ??? - Pn
CYCLIC DEFECT GROUPS, I
of p-subgroups Pi of G. Any such C has length < C < s n, initial subgroup P0 , and final subgroup Pn . Its conjugate by any element t g E is the p-chain C t : P0t - P1t - ??? - Pnt of G. Its normalizer in any subgroup H F E is the subgroup NH Ž C . s NH Ž P0 . l NH Ž P1 . l ??? l NH Ž Pn . . Clearly conjugation is an action of the group E on the family C Ž G . of all p-chains of G, and NE Ž C . is the stabilizer in E of the chain C g C Ž G . under this action. We denote by P Ž G ¬ 1. the family of all p-chains C of G whose initial subgroups P0 are 1, and by E Ž G ¬ 1. the subfamily of all C g P Ž G ¬ 1. whose final subgroups Pn are elementary abelian. The p-chain C in Ž7.1. is radical in G if its initial subgroup P0 is the largest normal p-subgroup O p Ž G . of G, and its ith subgroup Pi is the largest normal p-subgroup O p ŽNG Ž Ci .. of the normalizer of its ith initial subchain Ci : P0 - ??? - Pi for each i s 0, 1, . . . , n. Each of the families P Ž G ¬ 1., E Ž G ¬ 1., and RŽ G . is an E-invariant subfamily F of C Ž G .. For any such F we denote by FrG an arbitrarily chosen family of representatives for the G-conjugacy classes in F. If C is a p-chain of G, then NG Ž C . s NE Ž C . l G is a normal subgroup of NE Ž C .. We denote by NF Ž C . the image « ŽNE Ž C .. of NE Ž C . in F. Then Ž3.2. restricts to an exact sequence 6
NE Ž C .
NF Ž C . ª 1
1 ª NG Ž C .
of finite groups. The group NE Ž C . acts by conjugation Ž2.8. on the set BlkŽ O wNG Ž C .x. of all p-blocks b of A wNG Ž C .x. Its normal subgroup NG Ž C . fixes each such b. As in Ž2.9., we can use the epimorphism « in Ž7.2. to transfer this action of NE Ž C . to one of NF Ž C . on BlkŽ O wNG Ž C .x.. We denote by NE Ž C, b . and NF Ž C, b . the stabilizers in NE Ž C . and NF Ž C ., respectively, of any b g BlkŽ O wNG Ž C .x.. Then Ž7.2. restricts to the exact sequence NE Ž C, b .
NF Ž C, b . ª 1
1 ª NG Ž C .
of finite groups. In a similar way NE Ž C . acts by conjugation on the set IrrŽ A wNG Ž C .x. of all irreducible F-characters c of A wNG Ž C .x, with NG Ž C . fixing each such c . We use the epimorphism « in Ž7.2. to transfer this action to one of
EVERETT C. DADE
NF Ž C . on IrrŽ A wNG Ž C .x.. We denote by NE Ž C, c . and NF Ž C, c . the stabilizers in NE Ž C . and NF Ž C ., respectively, of any given c g IrrŽ A wNG Ž C .x.. Then Ž7.2. restricts to an exact sequence NE Ž C, c .
NF Ž C, c . ª 1
1 ª NG Ž C .
of finite groups. As in Ž3.12., we have Ž7.5. The exact sequence Ž7.4. is a subsequence of Ž7.3. whene¨ er c g IrrŽ b .. In addition to the usual p-block B of A w G x, we now fix a subgroup I of F and an integer d. We know from w3, 10.14x Žor, at least, we would know if the misprinted A wNG Ž P .x in the statement of that proposition were replaced by the correct A wNG Ž C .x. that any p-block b of A wNG Ž C .x induces a p-block b A wG x of A w G x. The following definition is essentially w4, 4.6x. DEFINITION 7.6. We denote by kŽ C, B, d, I . the number of characters c satisfying
c g Irr Ž A NG Ž C . d Ž c . s d,
BŽ c .
NF Ž C, c . s I.
Obviously kŽ C, B, d, I . depends only on the normalizer NE Ž C . of C and not on C itself. Furthermore, it remains constant when C is replaced by any G-conjugate C t . In view of w3, 1.17x this implies that Ž7.8.
If O p Ž G . s 1, then the ¨ alue of the alternating sum < < Ý Ž y1. C k Ž C, B, d, I .
is independent of the choice of F among the families P Ž G ¬ 1., E Ž G ¬ 1., and RŽ G . of p-chains of G. The invariant projective form w4, 4.7x of the conjectures in w2x, w3x can now be stated as Conjecture 7.9. If O p Ž G . s 1 and dŽ B . ) 0, then < < Ý Ž y1. C k Ž C, B, d, I . s 0
whenever F is one of the families P Ž G ¬ 1., E Ž G ¬ 1. or RŽ G .. Of course the conclusion Ž7.10. of this conjecture is independent of the choice of F by Ž7.8..
CYCLIC DEFECT GROUPS, I
Before stating the following theorem, which is the goal of this paper, we recall all its hypotheses. We have fixed a prime p and ground rings R, F, and F satisfying Ž2.1.. We have also fixed an epimorphism « : E ¸ F of finite groups with kernel G, and a totally split twisted group algebra A of E over F. We choose a p-block B of A w G x and a defect group D F G of B. We also choose an integer d and a subgroup I of F. Then we have THEOREM 7.11.
If D is cyclic, then Conjecture 7.9 holds.
Proof. Conjecture 7.9 holds trivially when D s 1, since B has defect dŽ B . s 0 in that case. It also holds trivially when O p Ž G . ) 1. So we may assume that D)1
O p Ž G . s 1.
We may also assume that the family F in Ž7.10. is E Ž G ¬ 1.. Suppose that kŽ C, B, d, I . / 0 for some p-chain C g E Ž G ¬ 1.. Then there is some character c g IrrŽ A wNG Ž C .x. satisfying Ž7.7.. The final subgroup Pn in C is elementary abelian, and hence normalizes all the other subgroups in C. It follows that Pn is a normal p-subgroup of NG Ž C .. By w3, 9.8x this implies that Pn is contained in any defect group DX of the p-block BŽ c . of A wNG Ž C .x. Since BŽ c . induces B by Ž7.7., its defect group DX is contained in some defect group of B, i.e., in some G-conjugate Dt of D Žsee w3, 10.12x.. Therefore Pn is an elementary abelian subgroup of Dt . Because D is a non-trivial cyclic p-group, this means that Pn is either 1 or the subgroup V Ž D .t of order p in Dt . It follows that C is G-conjugate to exactly one of the two p-chains C0 : 1
C1 : 1 - V Ž D .
whenever kŽ C, B, d, I . / 0. Hence Ž7.10. is equivalent to the equation k Ž C0 , B, d, I . y k Ž C1 , B, d, I . s 0.
The normalizers NG Ž C0 . and NE Ž C0 . are obviously G and E, respectively. The only possible p-block of A w G x inducing the p-block B of A w G x is B itself. So kŽ C0 , B, d, I . is the number kŽ B, d, I . of characters x g IrrŽ B . with defect dŽ x . s d such that NF Ž x . s « ŽNE Ž x .. is equal to I. ˜ s V Ž D . satisfies Ž3.6.. Since D is cyclic and non-trivial, its subgroup D ˜ E, ˜ and F˜ of D˜ Thus all the results in Section 3 hold for the normalizers G, in G, E, and F, respectively. The only non-trivial subgroup in C1 is ˜ s V Ž D .. So the normalizers NG Ž C1 ., NE Ž C1 ., and NF Ž C1 . are G, ˜ E, ˜ and D ˜ respectively. We know from Proposition 3.9 that there is exactly one F, ˜x inducing B. This and Definition 7.6 imply that p-block B˜ of A w G ˜ d, I . of characters x˜ g IrrŽ B˜. with defect kŽ C1 , B, d, I . is the number kŽ B,
EVERETT C. DADE
dŽ x ˜ . s d such that the image NF˜Ž x˜ . s « ŽNE˜Ž x˜ .. is equal to I. We conclude from this and the preceding paragraph that Ž7.12., and hence Ž7.10., is equivalent to
˜ d, I . . k Ž B, d, I . s k Ž B,
We denote by a the defect dŽ B . of the block B. Then p a is the order < D < of the defect group D. All the hypotheses of Theorem 6.1 are satisfied in our present situation. That theorem tells us that every character x g IrrŽ B . has height hŽ x . s 0, and hence defect dŽ x . s dŽ B . y hŽ x . s a. Since B˜ has the same defect group D as B, it has the same defect dŽ B˜. s a. So Theorem 6.1 also tells us that every character x ˜ g IrrŽ B˜. has ˜ Ž . Ž . Ž . Ž . height h x ˜ s 0 and defect d x˜ s d B y h x˜ s a. It follows that both sides of Ž7.13. are zero when d / a. Hence both that equation and Ž7.10. hold trivially in that case. Furthermore, these equations for the remaining case d s a are equivalent to
˜ I., k Ž B, I . s k Ž B,
where kŽ B, I . is the number of x g IrrŽ B . such that NF Ž x . s I, and ˜ I . is the number of x˜ g IrrŽ B˜. such that NF˜Ž x˜ . s I. kŽ B, The stabilizer NF Ž x . of any x g IrrŽ B . is contained in the stabilizer NF Ž B . of B by Ž3.12.. Hence we have NF Ž x . s N N F Ž B . Ž x . .
Similarly the stabilizer NF˜Ž x ˜ . of any x˜ g IrrŽ B˜. is contained in NF˜Ž B˜. by ˜ Ž3.15.. Since NF˜Ž B . is equal to NF Ž B . by Proposition 3.16, we conclude that NF˜ Ž x ˜ . s NN F Ž B . Ž x˜ . .
Theorem 6.1 gives us an isomorphism G of IrrŽ B . onto IrrŽ B˜. as NF Ž B .-sets. In view of Ž7.16. and Ž7.15. we have NF˜ Ž G Ž x . . s NN F Ž B . Ž G Ž x . . s NN F Ž B . Ž x . s NF Ž x . for any x g IrrŽ B .. So G restricts to a bijection of the set IrrŽ B, I . of all ˜ I . of all x˜ g IrrŽ B˜. with x g IrrŽ B . with NF Ž x . s I onto the set IrrŽ B, . NF˜Ž x s I. Hence ˜
˜ I . < s k Ž B, ˜ I.. k Ž B, I . s
CYCLIC DEFECT GROUPS, I
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