Variations on McKay's character degree conjecture

K and G/M is abelian, then we may employ the inductive hypothesis (with E replacing K) to conclude that H extends to G if and only if /j extends to N. We thus assume that O,.(G/K)= 1 or 1K1is odd. If lKl is even, then M/K is solvable by the Odd Order Theorem and G/M is abelian by the hypotheses of this theorem. The last two paragraphs imply that M/K is an elementary abelian y-group for a prime q and that M/K is a faithful irreducible G/M-module. Since G/M is abelian, G/M is in fact cyclic. Thus V/C z VK/Kz G/M is cyclic, whence /? extends to V and 9 extends to VK (see Lemma 1.4). It follows from the first paragraph that we are done in this case. Hence we assume that K is solvable of odd order. Let H = [K, P] a G and let H/L be a chief factor of G. Then P acts trivially on K/H, but not on H/L. Since H/L is abelian, C,:,(P) = 1. Let i. E Irr(H) be a constituent of 8,. By [6, Exercises 13.13 and 13.41, i. is P-invariant and I,, has a unique P-invariant irreducible constituent cp. By Lemma 1.2, 3.p(H, P) = cpp(L,P). Let iY= LN so that HU= G and H n U = L. Note that Kn U = LC d U. By Lemmas 1.2 and 1.3, (l,,c has a unique P-invariant irreducible constituent 7 such that [a,,, 21 is odd, dp(K, P) =yp(LC, P), and [yL, cp] ~0. Employing the inductive hypothesis, it suffices to show that 0 extends to G if and only if ‘/ extends to u. Let I= I,(;.). Since i.p(H, P) = cpp(L,P), it follows that In li= Z,.(q) 3 LP. If 0 extends to 6 E Irr(G), then bKIe Irr(KZJ 0) and Sz, is irreducible by the Clifford correspondence. This can happen only if KZ= G and conscquently LC(Zn V) = li. Similarly, if */ extends to I;, then LC(Zn U) = U and KZ=G. Thus, we do assume that KZ=G and LC(Zn U)= U. (See Fig. 1.) Choose pETrr(ZnKlE.) such that ~.?=6 and vEIrr(ZnKn (ily) such that vLc‘=y. Both p and v are P-invariant and pp(Zn Kj P) = vp(Zn Kn U / P) by Lemma 1.3. Using Clifford correspondence and Exercise 5.2 of [6], we see that character induction defines bijections from Irr(Zl p) onto Irr(GI 0) and from Irr(Zn I/‘] v) onto Irr( lil y). Since IG: II = IK: KnZj = (U: In UI = ILC: KnZn c’[, it follows that 8 extends to G if and only if ~1extends to Z and that “Jextends to U if and only if v extends to In li. If I< G, then the inductive hypothesis implies that Zi extends to Z if and only if v extends to In U. Hence we assume that I= Z,(J) = G and cpis invariant in U. If U = Z,(q), then cp” = A, ;rK= 0, and 4 + r” is a bijection from Irr(UI;1) onto Irr(GI 0). Since (G: Lii = IK: LCI, B extends to G if and only if ;’ extends to U. Thus we assume that li< Z,(q) and hence Z,(y) = G = Z,(O). If i, = cp, it follows from [S, Lemma 10.53 that whenever H d T< G.

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5 -+ rl.(, U is a bijection from Irr( 7’1;+) onto Irr( Tn U( ~0). In particular, t),, = y. Furthermore, 0 extends to G if and only if 7 extends to U. We arc done in this case. Since IKyl is odd, the chief factor H/L of G is abelian. Applying [S: Theorem 6.183, we may assume that 1, = etp, where e* = IH: LI. Note that all complements to H/L in G/L are normalizers of Hall-n-subgroups of G/L and thus conjugate, to U/L. Lemma 1.1 applies and let YE Char(G/H) be as prescribed in that Lemma. Since [OLc, 71 lK/ is odd, we have that t91.c= Yy,,.;‘. Furthermore rU= YV,a defines a bijection between r E Irr(G 12) and aE Irr( U I cp). If a extends 7, then 5,,c= Y,.,.y. Since every irreducible constituent of ‘sK lies in Irr(Kli.), it follows that rK= 8. Conversely, WCsee by the same argument that if r extends 0, then also a extends 7. 1. 1.6 COROLLARY. the canonical

i. E Irr(M/K) (i) (ii) (iii) (iv)

Assume the hypotheses of Theorem 1.5. Let t? and /? be extensions of x and /I to M and CP, respectively. Let be G-invariant and linear. The following are equivalent :

0 extends to G; ,. 0 extends to G; i-8 extends to G;

(v)

j? extends to N; B extends to N;

(vi)

&.,fi

extends to N.

Proof Since 2 is G-invariant, and (IG: MI, ]M: KI) = 1, i. extends to a linear character 11~Irr(G/K). Hence (ii) implies (iii). Clearly (iii) implies (i). Suppose 0 extends to XE Irr(G). Then x,,,= Sfi for a linear and G-invariant 6 E Irr(MjK). As above, 6 extends to G. If y is the complex con-

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jugate of an extension of 6 to G, then 7~ extends 0. Hence (i), (ii), and (iii) are equivalent. Similarly (iv), (v), and (vi) are equivalent. By Theorem 1.5. (i) and (iv) are equivalent. 1 1.7 COROLLARY. Assume that G is n-sepurahle, K is u normal n’-subgroup of G, MjK is a normal z-subgroup of G/K and PE Hall,(M). Suppose that t3E Irr( K) is P- invariant and /3 = Bp(K, P) E Irr(C,( P)). Jf G/M is abelian or 1KI is odd, then 0 extends to G !fi fi extends to NJ P). Proof: By the Frattini argument KN,(P) = G. Let VE Hall,.(N,(P)) and note KVE Hall,.(G). By Lemma 1.4, 0 extends to G if and only if B extends to KV. Similarly, /I extends to N,(P) if and only if / extends to V. Since V/C,(P) 2 VK,fK is isomorphic to a subgroup of G/M, we may without loss of generality assume that G = MV and N = PV. Then (IG,‘MI, IM/K[)= 1. Apply Theorem 1.5. 1

Let G be z-separable. Then B,(G) denotes a subset of Irr(G) that includes all z-special characters of G. We refer the reader to Isaacs [6] for definitions and properties of B,-characters. We do mention that B,-characters behave well when restricted to normal subgroups and also that restriction to n-elements gives a bijection from B,(G) onto Z,.(G) (see Lemma 1.8 below). For convenience, with no loss of generality we usually concern ourselves with I,,(G) instead of Z,(G). For v]E Char(G), we let q” be the restriction of r~to the z-elements of G and note that q” is a positive Z-linear combination of I,,-characters of G. Since each YE I,.(G) can be lifted to a BE-character of G (a stronger statement follows), we have that q0 restricted to a subgroup J of G is in fact a positive Z-linear combination of I,. characters of J. For N s G with OEB,(N) and ~EI,.(N), WClet B,(GIO)= Irr(G10)n B,(G) and I,,(Glq) denote those PE 1,.(G) such that 43is a constituent of p,,,. 1.8 LEMMA. Let Ng G with G n-separable und let 0 E B,(N). HoE I,.(N) and x + f is a hijection jkom B,(G IO) onto I,.(G IO”). Prooj:

Then

This is immediate from Isaacs [7, Corollaries 10.2 and 7.51. 1

If H ,< G and p E I,.(H), we may define a class function p(; defined on the n-elements of G with degree P’(‘(1) = IG: Hj p( 1). It is a consequence of Lemma 1.8 and remarks preceding it that pG is a positive Z-linear combination of elements of I,.(G). If H_a G, then repeated application of Lemma 1.8 and [7, Theorem 7.11 show that Z,,(G I p) is the set of irreducible constituents of pG‘. WC give a Clifford theorem for Brauer characters, but first quote Isaacs’ generalization (Theorem 1.9) of a result of Gallagher (Exercise 11.10 of [6]). Suppose that G is r-separable, N 9 G, and 43E I,.(N) is G-invariant. Let

130

THOMAS R. WOLF

g E G, and choose an extension i E I,.( (N, g)) of 9 (one does exist). We say that Ng is a q-special coset if 1.is invariant in C, where C/N = C,,,v( g). This is independent of the choice of i and the choice of the coset representative. 1.9 THEOREM. Assume that G is n-sepurahle or that n’ = { p ). Let N _a G and 43E Z,.(N) he G-invariant. Then 1I,,(G I cp)( equals the number of cp-special classes of n-elements of G/N. Proof:

This is [S, Theorem 6.21. 1

In light of Lemma 1.8, the above theorem may be restated for n-separable G in terms of B,-characters. Also, if rr contains all prime divisors of G, this yields Gallagher’s Theorem. 1.10 LEMMA. Let N be u normal subgroup qf‘ a x-separable group G, let 0 E B,(N), and let I= IG(0). Then Y -+ Y” dejines a bijection from I,,(/[ 0’) onto I,,( G IO’). Proof‘: By Induction on jG/N(. We first prove this when G/N is a z-group or rc’-group. In these particular cases, the result follows once we establish that r] + q’ is a bijection between B,(ZI 0) and B,(G IO). If G/N is a r-group this follows from the usual Clifford correspondence and [7, Theorem 7.11. If G/N is a &-group, both B,(G 10) and B,(II 0) are singletons [7, Theorem 6.21, say B,(II 0) = { yl}. Of course qG‘E Irr(G 10) and by [7, Corollary 6.41, qG‘E B,(G). We are done if G/N is a n-group or &-group. Choose N < M 4 G such that G/M is a n-group or a’-group. By the induction hypothesis, /I + 11” is a bijection from Z,,(ln M 10’) onto I,.(M 10’). Note that this map commutes with conjugation by 1, and that for each /))EI,,(ln M ( O’), we have I#“‘) d MI. Also, for 6 E Z,.(G loo), the distinct irreducible constituents of 6, that lie in I,,( MI 1)‘) form exactly one I-orbit. Consequently it suffices to fix /I E I,,( M n I( 0°) and show that Y + Yy” defines a bijection from I,,(11 8) onto I,(G 1b”). Let J=ZG(/IM) so that JdMZ and Jn I=I,(/?). If J< G, the inductive hypothesis implies that 7 + rJ defmes a bijection from I,.(ZnJlO’) onto Z,,(JIO”). Since M(I n J) = J, we have (rJ)M = z:,,, and hence r E I,.(In JI p) if and only if rJe r,.(,‘“). Hence, if J< G, then T + tJ is a bijcction from Z,.(Jn 11/I) onto I,,(JI /I”). By the first paragraph, character induction yields a bijection from Z,.(JI b”) onto I,.(GIfi”) and a bijection from /,,(JnIl~) onto I,,(rl/I). It follows that character induction is a bijection from I,,(11 fi) onto I,,(G I fi”). Thus we may assume that J= G. In particular, G = MI. Let x E /,.(G I /i,M). Then x = p” for some p E B,(G IQ). In particular,

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,U= <” for some
x E BAG I W. Proof: By Induction on IG: NI. Let M/N be a chief factor of G. First assume that M/N is a n-group. Let 43E Trr(M) 0) and note cpE B,(M) and cp*o B,(M*). Let Z=Z,(cp). Then I* = Z,,(cp*) and (I, M, cp)z (I*, M*, cp*). By the inductive hypothesis, there is a bijection between B,(ZI cp) and B,(Z* 1q*) that respects degrees.Employing Lemmas 1.8 and 1.10, we conclude that there is a degree-respecting bijection between B,(G( cp) and B,(G* 1cp*) for every 43E Irr(MI 0). Since (q*)“’ = (cp”)* for each KE G, it follows that there is a degree-respecting bijection between B,(G IfI) and B,(G* IO*) when M/N is a z-group. We thus assume M/N is a rr’-group. Now B,(MlO) and B,(M* IO*) are singletons, say {p} and {v}, respectively. Then p and v are invariant in G and G*, p extends 0, and v extends H*. Since B,(G I pu)= B,( G I 0) and B,( G* j v) = B,( G* I e*), we can conclude the proof by invoking the inductive hypothesis once it is established that (G, M, p) z (G*, M*, v). Since p* and v are G*-invariant extensions of 8*, p* = j,v for a G*-invariant linear j. E Irr(M*/N*). Since p* and v extend to S whenever S/M* E Hall,.(G*/M*), then i. also extends to S. Since O(i.) is a n’-number, it follows from Lemma 1.4 that 1.extends to G. Let r be the complex conjugate of an extension of 3. to G. Then (i, G): (G*, N*, f?*) =+ (G*. N*, 0*) is a character triple isomorphism, where i: G*/N* -+ G*IN* a(T)=r,f: is the identity and for N* < U d G* and ZECh(L’\e*), In particular, as z,~P* = v, we have that (G, M. p) s (G*, M*, p*) z (G*, M*, v).

1

1.12 PROPOSITION. Let (G, L, X) he a character triple whew G is cxE B,(L). (a) Then there exists a charactw triple

7c-separable and

132

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R. WOLF

(G*, L*, CX*) isomorphic to (G, L, a) such that a* is linear and L* is a z-group. In particulur, L* < Z(G*) and r E B,(L*). (b) Assume, in addition, that there exist L < T < G and T E Irr( T 1a) such thut (G: TI and T( 1)/r( 1) are o’-numbers, then L* may be chosen to be an o’-group. Proof: First observe that there is a character triple (G,, L, , r,) isomorphic to (G, L, x) with 3, linear and faithful and L, L. Note that C = N n K < N and KN = G. The Glauberman-lsaacs correspondence p(K, P) yields a bijcction between Irr,(Kl a) onto Irr,(CI a) and Up(K, P) is a constituent of 0, whenever 0~ Irr,(K( 2). If, in addition, 0~ B,(K), it follows, as K/L is a n-group or &-group, that every irreducible constituent of I), is a B,-character by [7, Corollary 6.6 and Theorem 7.11 and hence

MCKAY’S

CHARACTER

DEGREE

CONJECTURE

133

Op(K, P) E B,(Cl z). If K/L is a x’-group, then both B,(Ki ~1)and B,(CI r) are singletons by [7, Theorem 6.21. If K/L is a n-group, then Irr(KI 3) = B,(Kjx) and Irr(CIcr)=B,(CIcr) by [7, Theorem7.11. Hence p(K,P) maps Irr,( K 1r) n B,(K) onto Irr,(CI u) n B,(C) in a one-to-one manner. If 2 is a B,-character of G (or of N, respectively) with w’-degree, then xx (or xc, respectively) has a P-invariant irreducible constituent. Since p(K, Pj commutes with conjugation by N and since G = KN, it is sufficient to fix 0~B,(Klr) and /?=Op(K,P), and to show I(xEB,(Glf))l~(l) is an e,‘-number}1 = I{ YE B,(NIfl)l !P(l) is an w’-number}l. Let 0 and fi be the canonical extensions of H and /I to KP and CP, respectively. Note that t?and /? are B,-characters. For linear E.E Irr(KP,lK) with kernel R, both i-0 and fi extend 8, E B,(R). Hence A?E B,(KP) if and only if O(jb) is a n-number. In particular, A?E B,(KP) if and only if jL(pfl E B,(CP). If 2 E Irr(G 10) has w’-degree, the constituents of xKP are necessarily of the form 60 for linear 6. Thus it sufhces, as G = KN, to fix a linear 1.g Ir$KP/K) with O(i.) a x-number and to show IB,(GIifi)l = IS,(NI i,,p)l. If I= ZG.(jb),observe that I = I,(r.O), In N = I,+.(/,,:,) = l,V(&.pp), and I= K(ln N). By Theorem 1.9, it suffices to show that ne extends to U if and only if $ extends to Gin N whenever U/KP is abelian. In particular, we may assume that i is invariant in C;. We finish by applying Corollary 1.6. [ 1.14 THEOKEM.

Let G he x-separable und c+sepamhle. Let L 9 G, and N/L = N,;,( P/L). Suppose that cpE I,.(L) i.y .Y = {xc I,,(Gl cp)I x( 1) is an w’-number} and 9 = {/?~I,,(NIcp)l/1(1) is an w/-number}. Then I.vil = \.Fl.

P/L E Hall,(G/L), P-invariant. Let

Prooj: We argue by induction on IG : LI. Since P < I,(cp), we use the inductive hypothesis and Lemma 1.10 to assume that cpis G-invariant. Now let K/L = O”““(G/L). If K = L, then IV= G and the result is trivial. Choose a chief factor K/H of G such that L 6 H and then let J/H = N,:,(PH/H). Note that N d J < G and KJ= G. Now G/H acts on &( H I p) since Z,(Q) = G. Since (I PI, IK/HI ) = 1, it follows from Glauberman’s lemma (13.9 of [6]) that if r,, rz E.&(HI cp) are P-invariant and K-conjugate, then ~1,and !.xzare in fact D-conjugate where D,iH= C,,,(P). Since D d J and KJ= G, we may choose P-invariant p,, .... p, E .P,.(H I cp) such that each P-invariant slE .gx.(H I cp) is J-conjugate to exactly one p, and G-conjugate to exactly one pi. Set W = { Y E&(JI cp)l Y( 1) is an o’-number). If YES and 6 c.Y,,(H) is an irreducible constituent of Y,,, then IJ: I,(S)1 is a n’-number and consequently 6 is J-conjugate to one p,. It follows that the restriction of each x E Y u .% to H has exactly one pi as an irreducible constituent (not counting multiplicities). If L < If, the inductive hypothesis implies that I{ ): E &(G I p,) I ,Y(1) is a w’-number >i =

134

THOMAS R. WOLF

I{ YuEX,(./Jpi)l Y(1) is a o’-numbcr}( for each i Hence 19’1= I.%/. Since .I< G, the inductive hypothesis implies that 1.91= 19;) and so, if L < H, we have 19’1= IFI, as desired. We thus assume that L = H and hence that K/L is a chief factor of G and an of-group. We now apply Theorem 1.13 and Lemma 1.10. 1 1.15 COROLLARY.

Let G he n-separable and o-sepurable. Let L g G, und N/L = N,,,(P/L). If (I E B,(L) is P-invariant, is N o’-number}1 = I{!~‘EB,(NIU)I Y(1) is a then I~x~fUGI~~)Ix(1) o’-number } I. P/L E Hall,(GjL),

Proof:

Apply Theorem 1.14 and Lemma 1.8. 1

The next corollary is an unpublished result of Isaacs regarding IX,(G)l, where X,(G) denotes the set of x-special characters of G. 1.16 COROLLARY. P E Hall,(N,(H)). of P.

Let G be n-separable. Let HE Hall.(G) and let Then IX,(G)/ equals the number qf conjugucy classes

Proof: By [7, Corollary 5.43, X,(G) = { Y E B,(G) ) !P(1) is a n-number}. Applying Corollary 1.15 with L = 1 and o = rr’, we get that IX,(G)1 = JX,(N,(H))I. Since N,(H) has a normal n-complement H, then Pz N,(H)/H and X,(N,( H)) = Irr(N,(H)/H) by [7, Lemma 2.3 and Corollary 5.33. 1

At least when rr and o are singletons, say {p} and {y }, respectively, one may ask whether Theorem 1.14 generalizes to arbitrary G. The answer is no, as is evidenced when G = AS, p = 2, and q < 5. 2

The usual version of McKay’s conjecture for o-separable G is a corollary of Theorem 1.14. Actually, the original conjecture was just for p = 2 and ordinary characters of simple G. A more relined version of this conjecture is the Alperin-McKay conjecture which states that if B is a p-block of a group G with defect group D and if b is a block of NJ D) with b” = B, then b and B have the same number of ordinary characters of height zero. This was proven for p-solvable G in [lo]. An important aspect here is the relationship between Brauer induction and the Glauberman- Isaacs correspondence. We conclude by sketching a proof of this for both ordinary and Brauer characters of n-blocks of rr-separable groups. We also compare the number of Brauer characters of B and b. We deal with only one set n of primes in this section and thus use the following notation. If L d G and

MCKAY'S CHARACTER DEGREECOXJECTURE

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veI,(LL then 4GIcp)=IZ,(GIv)I and ~,(GIcp)=l{s~Z,(Glcp)l~(l)/~(l) is a rr’-number}l. Note the change in emphasis to 1,. Fong [4] provided reduction techniques for p-blocks of p-solvable groups. The analogous results for 7c-blocks appear in Slattcry [ 11, 123. If N is a normal n’-subgroup of a n-separable group G and if x E Irr(N), then Irr(G 1a) u I,(G I r) is a union of n-blocks of G. We denote this set of blocks by BE,(GI a). Let I= I,(r). There is a unique bEBI(II 2) such that B= { !PI Y’E h} and furthermore character induction is a height preserving map from b to B. Also B and b have a common defect group. We denote this by B= bf G, suppressing mention of x. This should not be confused with the Brauer correspondent b’ when defined. Finally, if I= G and N = O,,(G), then BI,(G Ix) is a singleton with a Hall-rr-subgroup as defect group. The following lemma is an immediate consequence of Section 3 of

c121. 2.1 LEMMA. Let G be n-sepurable. let L = O,.(G), and let r E Irr(L). Let I=Z,(a), let BEBI,(Gla) wirh defict group DcI. Let C=C,(D), N = N,;(D), und ,8 = rp(L, D) E Irr(C). Then (a) There is u unique 7c-block h oj’ N with bG‘= B. Furthermore D is a defect group for b;

(b) be WNI P); (c) [f B,EBl,(/la) with B=B,TG and if b,EBl,(/nNI/?) hh = B, then b,, t N = b (i.e., (b, 7 N)G‘ = (6;) r G). (d)

IfI=G,

An ordinary

then h=Irr(NI~)u/,(N(~) (Brauer) x E B hasheight

with

and DEH~IIJG). zero if x( 1),/IG: DI, = 1.

2.2 THEOREM. Let B be a x-block of a n-sepurable group G. Let D be a defect group for B and let b be the unique n-block of NG.(D) with bti = B. Then (i) The number of ordinary characters of height zero in B equals the number of‘ ordinary churacters of height zero in h; (ii) The number of height zero Brauer charucters in B equals the number of height zero Brauer characters in h. Proof Let L= O,.(G). WC may choose IXE Irr(L) such that BE BI,(G 12) and D s IG.(a). Let j3 = ap(L, D). Then /3E Irr( C), where C = C,(D) = NJ D) n L. Choose the unique B, E B1,( I,(r) ( r ) and b, E BI,( ZJa) n N,(D) ( ,6) such that B, 1 G = B and 6, r Z,,(r) = h. By comments preceding Lemma 2.1, B, and B have the same number of ordinary (Brauer) characters of height zero and also b, and h have equal numbers of height zero characters. dh,Iii 1.10

136

THOMAS

R. WOLF

Employing Lemma 2.1(c) and the inductive hypothesis, we may assume that a is G-invariant, B= Irr(G1 a) u /,(Gl z), b is invariant in N,(D), h = Irr(N,(D) 1a) u I,(N,(D) I CI),and DE Hall,(G). Observe that LN,( D)/L = N,,,,( LD/L). Let B, = Irr( LN,( D) ) a) u Z,(LN,;(D) 1x). Then B, is a n-block of LN,,(D) and !J’.~~(~)= B, by Lemma 2.1(d). By Theorem 1.14, B and B, have the same number of ordinary (Brauer) height zero characters. Thus we may assume that G = LN,(D). Two applications of Theorem 1.14 with appropriate choices for the sets of primes yield parts (i) and (ii). 1 2.3 THEOREM. Assume that L _a G, G is n-separable, P/L E Hall,(G/L), N/L = NG:L(P/L), and cpE I,(L) is P-incxriant. Then _ (a)

4GIcp)d4NIv);

Proo$ Note that part (c) follows from part (b) and Theorem 1.14. We prove (a) and (b) by induction on lG: LJ. By Lemmas 1.8 and 1.10, we may assume that cpis G-invariant. Let E/L = O,(G/L) and note that E c P E N and Nci:,<(P/E) = N/E. Also there is a unique TE I,( El cp),T extends cp,and z is G-invariant. If E > L, we finish by applying the inductive hypothesis. We thus assume that O,(G/L) = 1. Let K/L = O,.(G/L) # 1. By Proposition 1.12(a), we may assume L< Z(G) and that L and hence K are &-groups. Let M/K= O,(G/K)# 1 and Q/LEH~I~,(M/L) with QcP. Set J/L=N,(Q/L) and C=C,(Q)=KnJZL. Note that J
I(GIcp)31(Jlcp)31(Nl~)~[,(NIcp) and certainly part (a) follows. By two applications of Theorem 1.14,

[,(Glcp)=f,(JIcp)=I,(Nlcp). In particular, I(G 1cp)= I,(G 1cp) implies f(G 1cp)= I( N 1cp). Conversely, if WC must have f(JIq)=f(Nlcp)=f,(Jlcp) by the last paragraph and thus f(G 1cp)= /,(Gl cp).This establishes the claim. Note that CQ = Cx Q, for Q,,E HalI, and hence each by f,(Cl cp) is Q-invariant. By Lemma 1.2, there is a bijection p from (0 E I,( KI cp)10 is

f(G(cp)=f(Nlcp),

MCKAY'S CHARACTER DEGREE CONJECTURE

137

Q-invariant} onto Z,(Cl cp). Since Q 9 J, p commutes with conjugation by J. Since KJ= G and K n J= C, it suffices to fix a Q-invariant 0 E I,( KI 43) and /I = Bp and prove that I(G/ cp)= l(Ji p). (The possible existence 01 ~EZ,(KI cp) that are not Q-invariant lead to a possible inequality in l(GIcp)>l(Nlcp).) However, 1(G/cp)=/(G]b) is implied by Corollary 1.7 and Theorem 1.9. 1 2.4 FROPOS~TIO~. Let G be z-separable. Then each cpE Z,(G) n’-degree #‘and only if G has a normal Hall-rr-subgroup.

has

Proof Since O,(G) d ker(X) for all x E B,(G) z Irr(G), one direction easily follows. Now assume that p( 1) is a &-number for all p E I,(G). This property is inherited by normal subgroups and factor groups thereof. Arguing by induction on IG/, routine arguments show that G has a normaln-complement K and P E Hall,(G) fixes each p E Irr(K). Then P fixes each conjugacy class of K, by Brauer’s permutation lemma [6.32 of 61 applied to each element of P. Since ([PI, /KI) = 1, P centralizes K and Pa G. i

For II = {p}, the above proposition is valid without any solvability conditions. Known proofs used the classification of finite simple groups. We use it in the next corollary. 2.5 COROLLARY. Let G he 7c-separable, let P E Hall.(G) and N= N,(P). Then l(G) > l(N) with equality ifs P A G. Proof

Apply Theorem 2.3 with L = 1 and Proposition 2.4. 1

2.6 THEOREM. Let B be a n-block of a n-separable group G. Let D be u defect group of B and let b be the block of N,(D) satisfyjing b” = B. Then l(B) > l(b) and equality holds ff and only if l(B) = l,,(B). Note: If equality holds, we also have l(b) = l,(b). Proof: This follows from Lemma 2.1 and Theorem 2.3 with a proof analogous to that of Theorem 2.2. [

For p-blocks of finite groups, Alperin [I]

shows that the inequality

l(B) B l(b) would follow from his “weight conjecture,” and that for solvable

G, this conjecture follows from work of Okuyama. In particular, that approach offers a different proof of the above inequality for rc= {p) and G solvable. In [9], solvable groups Gi are constructed with L,= O,,(Gi) 6 Z(G,). r,~Irr(L~), Bi a block of G, such that Z,(Bi)=Z,(Gilzj)= {x,}, p [xi(l). and the p-length of Gi is i. Compare with Corollary 2.5.

138

THOMAS R. WOLF ACKNOWLEIXMENTS

Research supported by Deutsche Forschungsgemeinschaft and the Ohio University Research Council.

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