On Brauer's main theorem on blocks with normal defect groups

On Brauer's main theorem on blocks with normal defect groups

JOURNAL OF ALGEBRA 22, 1-l 1 (1972) On Brauer’s with WOLFGANG Mathematisches Main Theorem Normal HAMERNIK Institut, Defect on Blocks Groups AN...

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JOURNAL

OF ALGEBRA

22, 1-l 1 (1972)

On Brauer’s with WOLFGANG Mathematisches

Main Theorem

Normal HAMERNIK Institut,

Defect

on Blocks Groups

AND GERHARD MICHLER

Universittit

Tiibingen, West Germany

Communicated by J. A. Green Received July 22, 1970

1. INTRODUCTION Let F be a field of characteristic p > 0 dividing the order 1 G / of the finite group G. Since the block ideals B of the group algebra FG are in a one-to-one correspondence with the centrally primitive idempotents e of FG and with the (equivalence classes of) linear characters h of the center Z(FG) of FG a block of FG is denoted by B f-f e tt A. A character-free proof of the first main theorem on blocks operating completely within the center of the group algebra FG over an arbitrary field F of characteristic p > 0 has been given by Rosenberg in his paper [ll] and recently by Brauer [4] and Green [6]. It is the purpose of this note to give such a proof in the same generality for R. Brauer’s main theorem on blocks with normal defect groups: THEOREM 3.4. Let F be a field of characteristic p > 0 dividing the order / G j of the jinite group G. Let D be a p-subgroup of G, H = an,(D), K = D&-(D), R = H/D and i? = K/D.

Then there is a one-to-one correspondence g, between the blocks B tte tth of FG having defect group 6(B) =o D and the i7-conjugacy classes of blocks b ~ft)fi of FK having defect zero and covered only by blocks of FIl with defect zero. If F is a splitting field for K then Theorem 3.4 is equivalent to Theorem (12A) of Brauer [l] as is shown in Corollary 3.5. Our proof is inspired by Passman’s article [9]. In the course of the proof of Theorem 3.4 some other results of Brauer’s papers [l-3] are obtained and generalized for group algebras FG of finite groups G over arbitrary fields F of characteristic p > 0 using Rosenberg’s ring theoretical methods

[Ill. Copyright Q 1972by AcademicPress,Inc. All rights of reproductionin any form reserved.

1

2

HAMERNIK AND MICHLER

Concerning our terminology we refer to Rosenberg [ll], Curtis and Reiner [5], and Zariski and Samuel [12]. Our notation is that of [9] which is common.

2. BLOCKS AND NORMAL SUBGROUPS In this section G is a finite group and if not stated otherwise F is an arbitrary field of characteristic p > 0 dividing the order / G 1 of G. DEFINITION [9]. Let H be a normal subgroup of G and let b of* p be a block of the group algebra FH. Then T,(b) = {x E G / b” = b} is the inertia group of b oft) p in G. Clearly H < To(b). Let bj -fj t+ pcLj, j = I,..., t, be the distinct conjugates of b ++f + p. Then a = c=,fj is a central idempotent of FG. Therefore a = XI=, ei where each ei is a centrally primitive idempotent of the group algebra FG. Each block B, H e, t) Ai of FG covers each block b, t, fj f-) pj of FH. The usual definition of a linear character of the center Z(FG) of the group algebraFG (cf. [5, p. 6061) can easily be generalized for any finite dimensional F-algebra T. LEMMA 2.1. Let H be a normal subgroup of G. If B f-t e f-) A is a block and if b f+ f t, p is a block of FH then the restrictions A, and p1 of h and p to T = Z(FG) n Z(FH) are nontrivial linear characters of T.

qf FG,

Proof. T is a commutative artinian ring having the same identity element as FG and FH. Thus T $ ker X and T $ ker p. Therefore M = T n ker h < T. Since ker X is a maximal ideal of Z(FG) it follows that M is a maximal ideal of T because T is artinian. Clearly X,(T) g T/M which is a finite extension of F. Similarly it follows that p1 is a linear character of T. PROPOSITION 2.2. Let B f+ e f-f X be a block of FG and let b ++ f t, p be a block of FH where H is a normal subgroup of G. Then B covers b if and only if there exists an F-algebra isomorphism r from p(T) onto X(T) such that rrp(a) = X(a) for all a E T = Z(FG) n Z(FH). Proof. Let Z = Z(FG) and Z, = Z(FH). of 2 and let Jr be the Jacobson radical of Z, centrally primitive idempotents fi EFH, j = to f under G are fj = f zj where {xj E G I j T,(b) in G with x1 = 1.

Let J be the Jacobson radical . If 1 G : To(b)/ = t then the l,..., t, which are conjugate = l,..., t} is a transversal of

BLOCKS

WITH

NORMAL

DEFECT

We now assume that B covers 6. Then idempotents e = e, ,..., e, EFG such that

3

GROUPS

there are centrally

primitive

s=~f,=~e,tT=ZnZ~. j=l

i=l

If x were not a primitive idempotent of T then there would be two nonzero orthogonal idempotents y and x in T such that x = y + x. Since y, .z E Z, there are centrally primitive idempotents yn EFH, h = I,..., s, and z, EFH, q = I,..., u, such that y = xi=, yh and z = C:=, x, . Hence

x = glfj

= y + z = r, yh + i xg * 11=1 q=l

Since each central idempotent of FH is a unique sum of the (finitely many) centrally primitive idempotents of FH we may assume that f = fi = y1 . As (Z# = Z, for j = I,..., t we obtain Thus f =yl =yylEyZl. f, = fzj E ( YZ,)~~ = yz,Zzj = yZ, for all j because y E T < Z(FG). Hence x = yzr for some x1 E Z, . Because of x = &, fj E yz, . Therefore x = y + z = yzr it follows that x = x.z = ( yz& = .zr yz = 0, a contradiction! This shows that x is a primitive idempotent of the finite dimensional commutative F-algebra T. If J(T) d enotes the Jacobson radical of T then XT is a commutative local ring with maximal ideal M = xJ(T). Since XT is complete in the M-adic topology and since the characteristics of XT and of the field XT/M are equal to the characteristic p > 0 of F by I. S. Cohen’s structure theorem of complete local rings [12, p. 3041 there exists a field W contained in XT < T such that XT = W + x J( T) and W n x J( T) = 0. Thus for every a E T there exists a unique w E W and a unique n E x J( T) such that (*I

xa = w + n.

Since x = xi=, ei is the identity element of XT it follows from (*) and e, = e that ae = we + ne. Now 71E x J(T) < T implies 0 = h(n) = h(ne) because X is the linear character of Z(FG) belonging to e. Hence (**I

h(a) = h(ae) = h(we) + h(ne) = X(w).

By Lemma 2.1 the restriction h, of h to of T. Therefore the restriction 01 of X to from W onto h(T) by (**). Since x = xi=, fi is the identity of XT the restriction /3 of p to W is an F-algebra such that ~(a) = p(w) = /3(w). Th ere f ore

T is a nontrivial linear character W is an F-algebra isomorphism it follows similarly from (*) that isomorphism from W onto p(T) it follows from (**) that z-p(a) =

4

HAMERNIK

AND

MICHLER

h(a) for each a E T = Z(FG) n Z(FH) w h ere n is the F-algebra isomorphism c&l from p(T) onto X(T). Conversely let n be an F-algebra isomorphism from p(T) onto h(T) such that rip(u) = A(a) for all a E T. Since x = &jj is a central idempotent of FG there are centrally primitive idempotents ei E FG, i = 1,. . . , Y, such that x = xi=, ei . Thus from X(x) = rp(x) = z-&&ji) = TV(~) = ~(1) = 1 and B H e tf X follows that e E {e, ,..., e,>. Hence B covers b. This completes the proof of Proposition 2.2. Using Proposition 2.2, Passman’s proof of Fong’s Theorem [9, Thenrem 21 can now be suitably modified in order to obtain the following more general result. THEOREM 2.3. Let H be a normal subgroup of G and let F be a $eld of characteristic p > 0. If the block B, t) e, tf A, of FG is one of maximal defect among the blocks Bi f-f ei H hi , i = 1,..., r, of FG covering the block bt, f t) p qf FH and if a = x:=1 ei = C%, q$, where q8 E F and where c, is the class sum of the conjugucy class C, of G then the following statements hold:

(a) There is a p-regular conjugucy class, C, say, of G such that C, C H, q1 # 0, and A,(&,> # 0. (b) Wi) 6 W,) ==G S(G) (4 Vi) Gc T,(b) (4 WV n H =G S(b) (4 4 fW%)l) < 4 T&)1) (f) If F is a spZitting field for H then equality holds in (e). Proof. For the proof of (a), (c), and (e) we refer to the proof of [9, Theorem 21 which carries over verbatim to the case of an arbitrary field F. By Proposition 2.2 there are F-algebra isomorphisms ZT~from p(T) onto hi(T) such that p(y) = n;‘hi( y) for all y E T = Z(FG) n Z(FH). Hence 0 # Ai = ail = ~,r;‘h~(C,) and therefore Ai # 0 for i = I,..., Y. This implies S(BJ & S(C,) and from q1 # 0 follows S(C,) <, S(BiO) for at least one i, E {l,..., r}. Thus S(C,) & S(BiJ Gc S(C,) and 6(&o) =G S(C,). Since d(Bi,) < d(B,) it follows that S(B,) =c S(C,) and (b) is proved. Let x E C, and let c = xH. Let {xi E G j i = l,..., m} be a system of representatives of the left cosets of an,(c) in G. If c” = x:,,,g then

By (a) we have A,(& # 0 and by Proposition 2.2 there is an F-algebra isomorphism or from p(T) onto h,(T) such that rr+( y) = A,( y) for all y E T.

BLOCKS WITH NORMAL DEFECT GROUPS

Therefore

(*) Hence ~(&‘) # 0 for some j~{l,..., WZ}and the proof of (d) can now be completed by the same arguments as in [9, Theorem 21. If F is a splitting field for H then the F-algebra isomorphism n of Proposition 2.2 induces the identity on p(T) for every linear character p of Z(FH). Furthermore, &?) = pg-l(x) for all z E Z(FH) and all g E G. Thus from (*) we obtain CT=, pzi(e) # 0 and now by the same reasoning as in [9, Theorem 21 the proof is completed. For the statement and the proof of the following results we need DEFINITION 2.4 [3, p. 11181. Let U be a subgroup of G and let b f-f f t, p be a block of the group algebra FU. Let ~~(&~g) = ,~(x~.~.,~g) for all conjugacy classes C of G, where p(x sscnug)=OifCnU= ~.Ifthere is a block B o e t) X of FG such that pG and X are in the same equivalence class of linear characters of the center Z(FG) of FG then bG is defined and bG = B.

Remark. Immediately from the definition follows (cf. [2], (2C)): Let U, and U, be subgroups of G with U, < U, and let b, f-) fi t+ p1 be a block of FU, . If bp and bIG are defined then also (bp)G is defined and (bp)o = b,o. PROPOSITION 2.5. Let F be a field of characteristic p > 0. Let D be a normal p-subgroup of G and let K be a normal subgroup of G containing 5&(D). Then the following statements hold:

(a) bG exists for each block b f--f f tf TVof FK and it is the unique block B t, e f--f h of FG covering b. (b) For each defect group S(b) of b tf f tt TV there is a defect group S(B) of B t) e t) X such that S(b) = S(B) n K. (c) If F is a splitting field for K then / S(B) : S(b)1 = p* where r = ~(1 To(b) : K I). Proof. (a) It is easy to see that b is covered by at least one block B f-f e f-) X of FG. Thus Proposition 2.2 asserts that there exists an F-algebra isomorphism r from p(T) onto h(T) such that rrp(a) = A(a) for every a E T = Z(FG) n Z(FK). Therefore r&f?) = A(&‘) for every conjugacy class C of G contained in K. Suppose that pG is defined as in Definition 2.4. Then for every conjugacy class C of G we have if if

CgK C
6

HAMERNIK

AND

MICHLER

Hence pG is a linear character of Z(FG) which is equivalent to h because h(C) = 0 for every conjugacy class C of G such that C $ K. This is seen as follows. Since D is a normal p-subgroup of G each defect group 6(B) of B tf e et /\ contains D by Proposition 4.4 of [ll]. Hence !&(6(B)) < I&(D) < K and S(B) so 6(C) f or every defect group 6(C) of C. Thus h(C) = 0 by Lemma 3.1 of [II]. (b) follows immediately from (a) and Theorem 2.3(d). (c) If F is a splitting field for K then Theorem 2.3(f) asserts that

~(1S(B) : S(b)l) = v ( / S/;;zk, This completes

) = v (#“)

the proof of Proposition

= ~(1 T,(b) : K I).

2.5.

Remark. Proposition 2.5 contains as a special case Brauer’s Theorem (2F) of [2]. Another corollary of Proposition 2.5 is the following generalization of the first main theorem on blocks which essentially was proved by Brauer [3, Theorem 5C] in case F is a splitting field for G. THEOREM 2.6. Let F be a field of characteristic p > 0 dividing the order 1G 1 of the $nite group G. Let D be a p-subgroup of G and let K be a normal subgroup of H = 5&(D) containing 9!,(D). Then there is a one-to-one correspondencez,bbetween the blocks B tt e e, h of FG with defectgroups S(B) =o D and the representatives b tt f t, p of the H-conjugacy classes of blocks of FK having the property that b t, f e, p is only covered by blocks of FH with defect ~(1 D I). The correspondence9 is given by B = bG.

Proof. By the first main theorem on blocks [ll, Theorem 5.31 we may assume that D is normal in G, i.e., G = H. Let B e, e et h be a block of FG with defect group S(B) = D. Then there exists exactly one G-conjugacy class of blocks b t) f f-t p of FK covered by B. If b is a representative of such a G-conjugacy class of blocks of FK then B = bG by Proposition 2.5. Thus bt, f f+ p is only covered by blocks of FH with defect v(] D I). Suppose that b e, f t+ p is a block of FK which is only covered by blocks of FG with defect ~(1 D I). Then Proposition 2.5 asserts that B = bG is the only block of FG covering b. As D is normal in G every defect group S(B) of B contains D by Proposition 4.4 of [l 11. Therefore S(B) = D. Remark. More precisely we have to say that Brauer’s Theorem of [3] follows at once from Theorem 2.6 and the following lemma. LEMMA

(5C)

2.7. Let D be a p-subgroup of G and let K be a normal subgroup

BLOCKS WITH NORMAL DEFECT GROUPS

7

of H = ‘S,(D) containing Q,(D). Let F be a sphtting jeld for K. Then the following properties of the block b t) f c) p of FK are equivalent: (1) b H f t--f p is only covered by blocks of FH with defect ~(1D I) (2) s(b) =H D n K and ~(1 DK I> 2 r(l [email protected])l). Proof. Again we may assume that D is normal in G. Clearly (1) implies (2) by Theorem 2.3(d) and (f). Suppose that (2) holds. By Proposition 2.5(a) the block 6 f-f f tf p of FK is only covered by the block B = bo of PG. If 6(b) is a defect group of b then by Proposition 2.5(b) there is a defect group S(B) of B such that S(b) = 6(B) n K =G D n K. Since ~(1DK I) > ~(1 T,(b)l) Theorem 2.3(e) implies that ~(1DK I) = ~(16(B)K 1) = v(T,(b)). Hence 6(B) = D because D < 6(B) by Proposition 4.4 of [I 11. Remark. As the proof of Lemma 2.7 shows the implication from (2) to (1) even holds if F is an arbitrary field. As an application of Theorem 2.6 we state a generalization of Brauer’s criterion [2, Theorem 2E] for the existence of bc. COROLLARY 2.8. Let F be a field of characteristic p > 0 dividing the order / G j of the Jinite group G. Let D be a p-subgroup of G and let U be a subgroup of G containing f?,(D) and D. Then bC exists for every block b ++ f t+ p of the group algebra FU with defect groups S(b) =(I D.

Proof. If N = 5&(D) and K = D.&(D) then K is normal in N because e,(D) = !&-(D). Hence by Theorem 2.6 there exists a block 6, *fit) p1 of FK such that b = b,n. Now K is normal in ‘So(D) = H. Thus we may apply Theorem 2.6 again and obtain that bIG exists. But by the remark following Definition 2.4 brG = (6,U)G = bG. Hence bG exists.

3. PROOF OF THEOREM 3.4 By the first main theorem on blocks [l I, Theorem 5.31 we may assume that D is a normalp-subgroup of G. Since K = D!&(D) is a normal subgroup of G satisfying the hypothesis of Theorem 2.6, there is a one-to-one correspondence between th e blocks B t) et) h of FG having defect group D and the representatives b t) f f-f p of G-conjugacy classes of blocks of FK having the property that b of c-, p is only covered by blocks of FG with defect ~(1D I). Thus it remains to show PROPOSITION 3.1. Let D be a normal p-subgroup of G, K = D&(D), x = K/D, and C?= G/D. Then the canonical group algebra homomorphism

8

HAMERNIK

AND MICHLER

T from FG onto FG induces a one-to-one correspondence between the representatives b t) f t) p of G-conjugacy classes of blocks of FK and the representatives 6 wftt p of G-conj’ugacy classes of blocks of FK. Furthermore, ta) (b)

s(T(f)> 41 T&l

=R &cb)lD : E I) = 14 T,(b)

: K I)

(c) b t) f tf p is only covered by blocks of FG with defect ~(1 D I) zf and only if b+-, T(f) ++ p is only covered by blocks of FG with defect zero. The proof of Proposition 3.1 requires the following two lemmas due to Brauer [l] and Kawada [7, Section 71 which are included for the sake of clearness. LEMMA

3.2.

With the hypothesis of Proposition 3.1 the following

statements

hold: (a) p-regular classes of (b) (c)

7 induces a one-to-one correspondence between the class sums of the conjugacy classes of K and the class sums of the p-regular conjugacy R. Each p- reg u Iar .eonjugacy class C of K is contained in 9!,(D). 7(C) has defect group S(C)/D.

Proof. (b) follows immediately from [5, Lemma 40.31. The proofs of (a) and (c) are given by elementary group theoretical arguments [l, I lA]. LEMMA

3.3.

With the hypothesis of Proposition

3.1 the following

assertions

hold: (a) 7 induces a one-to-one correspondence between the blocks 6 t--f f e, TV of FK and the blocks 6 tt f tt ii of FK such thatf = 7(f) and S(b) == S(b)lD. (b) Let b, w,fi t, ,LQ and bj ~fj t, t~~ be blocks of FK and let x be an element of G. Then x-l&is = j;. if and only if x-lfix = f, . (4

T&Q

= T&)/D.

Proof [7, Section 71. (a) 7(f) # 0 by [ll, Lemma 4.21. Let fi ,..., fn be the centrally primitive idempotents of FK and let Jr ,. . . , jm be the centrally primitive idempotents of FK. Then T(I) = i = zF=,T(fi) = Cj”=,J;, . Let f ~(fr ,...,f,J. If T(f) were not a primitive idempotent of Z(FK) then & *fi t--f ,i& and 6, ~3~ t--f pa would be different blocks of FK satisfying ,$(~(f)) = 1 = ,&(T( f)). Since plT and ,&T are linear characters of Z(FK) We get p(e) s &(T(e))’ fOrj = 1,2 f or each p-regular conjugacy class C of K with class sum r?. Because of Lemma 3.2 this implies ,G,(e> 4 ,[email protected]) 1 p ^ ~$7 means that p and ,$T are in the same equivalence of Z(FK).

class of linear characters

BLOCKS

WITH

NORMAL

DEFECT

GROUPS

9

for all p-regular conjugacy classes L of K with class sum e. By a theorem of Osima [IO] each centrally primitive idempotent is a linear combination of p-regular class sums and therefore 1 = ,&(fr) e p,(fr) = 0, a contradiction! The second part of (a) follows immediately from Lemma 3.2 and the definition of a defect group. (b) (Due to J. A. Green). It is clear that X-~&X =f, follows from x~‘f~~ = fj . Conversely, if %-lfi~ = fj , then x-‘fi~ - fj E ker T n Z(FK). Since by Lemma 4.2 of [ll] ker 7 is nilpotent there is an integer c 2 0 such that (~-‘fi~ -f3)Yc = 0. Hence X-rfix = fj because X-‘fix and fj are block idempotents of FK. (c) follows immediately from (b). Proof of Proposition 3.1. (a) and (b) follow immediately from Lemma 3.3. Thus it remains to show (c). Suppose that b H f t, p is a block of FK with the property that b ++ f t, p is only covered by blocks B H e t) /\ of FG with defect d(B) = ~(1 D I). Since D is normal in G, Proposition 4.4 of [ll] implies that S(B) = D. Furthermore, B = bG by Proposition 2.5. If {gj E G /j = l,..., t} is a transversal of the inertia subgroup T,(b) = {g E G 1bg = b} in G then e = ~~=,fg~. Let & = g,D for j = l,..., t. Then Lemma 3.3 asserts that

(*)

-r(e) = i

7(f)Bj,

j=l

where {T( f)“j 1j = l,..., t} are all the G-conjugates of the block idempotent f = T(f) of FK. If u denotes the F-subalgebra of T = Z(FK) n Z(F(?) generated by the class sums of all p-regular conjugacy classes c ,< R of G with defect zero then I is a primitive idempotent of T contained in U by Proposition 2 of [8]. Thus T(e) E V where v is the F-subalgebra of Z(Ft?) generated by all class sums of the conjugacy classes C of G with defect zero. If T(e) = t?i + ... + E, where the ei are centrally primitive idempotents of Fe then ei = T(e)& E Y because ti is an ideal of Z(FG) by Lemma 2.2 of [I I]. Thus (*) asserts that 6t, T(f) f-) p is only covered by blocks of Fe with defect zero. Conversely, suppose that the block 6 ~ftt p of FK is only covered by blocks of F(? with defect zero. Then by Lemma 3.3(a) and (c) there exists exactly one block b t-f f t+ p ofFK such thatf = T(f) and T&6) = T,(b)/D. Therefore, if { gi E G 1j = I,..., t) is a transversal of T,(b) in G and gj = gjD forj = l,..., t, then

(**I

= i 7(f )“j E T = i=l

Z(FK)

n Z(FG)

10

HAMERNIK

AND

MICHLER

because{&Eclj = l,..., t} is a transversal of TG(~) in G. By Proposition 2.5, e = Cj”=,fgj is a centrally primitive idempotent of FG with defect group 8(e) 3 D. Hence it remains to show that 8(e) < D. By (**) T(e) = &,fgi E v r\ T because b++ft-t fi is only covered by blocks of Fe with defect zero. Since a = Y n T, T(e) E 0. Therefore T(e) is a linear combination of class sums Ci of p-regular conjugacy classes C< < E of G with defect zero, i = l,..., s say. Thus T(e) = zz=, riCi , ri E F. By Lemma 4 of [8] for every c$ , i = I,..., s, there is exactly one p-regular conjugacy class Ci of G with defect group S(Ci) = D such that T(CJ = Cc, and if ei is the class sum of Ci then .(ei) = ei. Hence T(e) = xi=, r,~(d~). Thus e = & rzei + U, where v E ker T. Since ker T is a nilpotent ideal by Lemma 4.2 of [II], and since Ci=, r,e, E Z(FG) it follows that

t***> for a suitable integer h 3 0. Clearly Ci=, rici •1~ , where ID is the Fsubalgebra of Z(FG) 5‘p anned by the class sums of conjugacy classes of FG with defect groups conjugate to (not necessarily proper) subgroups of D. By Lemma 2.2 of [II] I, is an ideal of Z(FG). Hence e E -To by (***), and 8(e) < D which completes the proof of Proposition 3.1. Remark. In Theorem 3.4 the correspondence v = TU, where (T denotes the Brauer homomorphism with respect to D, and where 7 denotes the canonical homomorphism of Proposition 3.1. Collecting our results we now can state the following corollary which also contains Theorem (12A) of Brauer [l] as a special case. COROLLARY

are equal

if F

3.5. With the notation of Theorem 3.4 the following is a splitting field for R:

(1) The number of all the blocks B t) e f-, A of FG with 6(b) =G D. (2) The number of all H-conjugacy classes of blocks b f--) f covered only by blocks of FH with defect ~(1 D I). (3) The numbs of all fi-conjugacy classes of blocks 6 wft, covmed only by blocks of Ff7 with defect zero. (4) The number of all i7-conjugacy classes of blocks 6 f-, 3 with defect zero and the property that p { 1 T&) : R I. (5) The number of all H-conjugacy classes of blocks 6 tf f with defect v(I D I) and the property that p r 1 T,(b) : K (. Proof.

The equivalence

of (I) and (2) holds by Theorem

numbers

defect group +S p of FK p of FK t+ p of FK ti

p of FK

2.6, and the

BLOCKS WITH

NORMAL DEFECT GROUPS

11

one of (1) and (3) is true by Theorem 3.4. Lemma 3.2 implies the equivalence of (4) and (5). Since D is a normal p-subgroup of K the field F is a splitting field for K if and only if F is a splitting field for K. Thus by Lemma 2.7 and Lemma 3.2 the statements (2) and (5) are equivalent. Finally we should like to mention that the results of Rosenberg and Passman to which we have referred are proved without using character theory. Moreover, we did not use any result of modular representation theory dealing with fields of characteristic zero. Added in proof: Using the results of this article the authors obtained by means of ring theoretical methods also R. Brauer’s third main theorem on blocks for group algebras over arbitrary fields (see [3]) asserting that under the hypothesis of Corollary 2.8 B = bc is the principal block of FG if and only if b tf f H p is the principal block of FU.

REFERENCES 1. R. BRAUER, Zur Darstellungstheorie

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

der Gruppen endlicher Ordnung I, Math. 2. 63 (1956), 406-444. der Gruppen endlicher Ordnung II, M&. Z. R. BRAUER, Zur Darstellungstheorie 72 (1959), 25-46. R. BRAUER, On blocks and sections in finite groups I, Amer. J. Math. 89 (1967), 1115-1135. R. BRAUER, On the first main theorem on blocks of characters of finite groups, Illinois 1. Math. 14 (1970), 183-187. Theory of Finite Groups and C. W. CURTIS AND I. REINER, “Representation Associative Algebras,” New York, 1962. J. A. GREEN, Axiomatic representation theory for finite groups, J. Pure Appl. Algebra 1 (1971), 41-77. Y. KAWADA, On blocks of group algebras of finite groups, Tokyo Kyoiku Daigaku A, 9 (1966), 87-110. G. MICHLER, Conjugacy classes and blocks of group algebras, Symposium on “Associative Algebras,” Rome, November 23-26, 1970. D. S. PASSMAN, Blocks and normal subgroups, J. Algebra 12 (1969), 569-575. D. S. PASSMAN, Central idempotents in group rings, Proc. Amer. Math. Sot. 22 (1969), 555-556. A. ROSENBERG, Blocks and centres of group algebras, Math. Z. 76 (1961), 209-216. 0. ZARISKI AND P. SAMUEL, Commutative Algebra II, Princeton, 1960.