Involutions, weights and p-local structure Geoffrey R. Robinson Institute of Mathematics, Fraser Noble Building, University of Aberdeen Aberdeen AB24 3UE, Scotland (e-mail: g.r.robinson@abdn.ac.uk)

Abstract

We prove that for an odd prime p, a finite group G with no element of order 2p has a p-block of defect zero if it has a non-Abelian Sylow p-subgroup or more than one conjugacy class of involutions. For p = 2, we prove similar results using elements of order 3 in place of involutions. We also illustrate (for an arbitrary prime p) that certain pairs (Q, y), with a p-regular element y and Q a maximal y-invariant p-subgroup, give rise to p-blocks of defect zero of N_{G}(Q)/Q and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J.L. Alperin.

Keywords: Block, Involution. AMS subject classification: 20C20

Introduction: Involutions have played a crucial role in finite group theory for many decades. They also figure prominently in representation theory, both ordinary and modular. Examples of the former include their occurrence in finite reflection groups, and an example of the latter is that in characteristic 2, J. Murray proved in [4] that the projective summands of the (characteristic 2) permutation module (under conjugation action) on the solutions of x^{2} = 1 in G are (in bijection with) the real 2-blocks of defect zero.

Involutions also influence representation theory in odd characteristic. It was proved by Brauer and Fowler in [1] that when p is an odd prime, then G has a p-block of defect zero if there is an involution t ∈ G which neither inverts nor centralizes any non-trivial p-element of G. This result was extended by T. Wada, who proved in [8] that if there are r mutually non-conjugate involutions of G which neither invert nor centralize any non-trivial p-element of G, then G has at least r distinct p-blocks of defect zero. We prove here that when p = 2, elements of order 3 can play a role analogous to that played when p is odd by involutions in the results above: we prove that the number of 2-blocks of defect zero of G is at least as great as the number of conjugacy classes of elements of order 3 which normalize no non-trivial 2-subgroup of G.

We also point out here that results of this nature can be combined with local group-theoretic analysis to prove that when p is an odd prime and G is a group without elements of order 2p, then G has a p-block of defect zero if it has more than one conjugacy class of involutions (we prove a more precise result without using the classification of finite simple groups, which could be sharpened even further by using that classification).

In a different direction, the celebrated weight conjecture of J.L. Alperin (in its non-blockwise version) defines (for a fixed prime p) a weight of G (up to conjugacy) as a pair (Q, S) where Q is a p-subgroup of G and S is an absolutely simple projective N_{G}(Q)/Q module in characteristic p. Alperin’s weight conjecture then asserts that the number of non-conjugate weights of G for p should be the number of conjugacy classes of p-regular elements of G (which is also the number of isomorphism types of absolutely simple modules for G in characteristic p). At present, there seems to be no reason to expect a natural bijection between weights and p-regular conjugacy classes, or between weights and characteristic p simple modules for G (though it is impossible to preclude the possibility that one or the other might emerge in future). Relatively few purely group-theoretic criteria are known to date which place non-conjectural bounds on the number of weights. We give some group-theoretic conditions of this nature which place lower bounds on the number of weights, using sharpenings of results of Brauer-Fowler [1], Tsushima [7] and Wada [8], going somewhat further than the present author’s results in [6], and incorporating the result about 2-blocks of defect zero and elements of order 3 which normalize no non-trivial 2-subgroup of G.

A naïve attempt at associating p-regular classes with weights of G might be to consider a p-regular element y and a maximal y-invariant p-subgroup Q. Then y normalizes no non-trivial p-subgroup of N_{G}(Q)/Q and it might be hoped that a p-block of defect zero of N_{G}(Q)/Q could be naturally associated to y (or yQ). More ambitiously, it might be hoped that weights could be parametrized in terms of conjugacy classes of pairs (Q, y) where y is a p-regular element of G and Q is a maximal y-invariant p-subgroup of G.

However, there are usually more conjugacy classes of such pairs (Q, y) than there are simple modules. The number of conjugacy classes of such pairs (Q, y) is equal to the number of simple modules precisely when C_{G}(y) transitively permutes the maximal y-invariant p-subgroups of G for each p-regular y ∈ G. In general, this need not be the case. For example, when p = 3 and G ≅ PSL(2, 11) we may take y to be an involution. There is a Sylow 3-subgroup Q of G which is centralized by y and there is another Sylow 3-subgroup R of G whose nonidentity elements are inverted by y. Clearly Q and R are not conjugate via an element of C_{G}(y).

We are nevertheless interested in pairs (Q, y), where y is p-regular and Q is a maximal y-invariant p-subgroup, and we will point out some instances where they give rise to weights.

Lemma 1: i) Let Q be a p-subgroup of G and y be a p-regular element of N_{G}(Q) such that yQ ∈ O_{p′}(N_{G}(Q)/Q). Then Q is a maximal y-invariant p-subgroup of G if and only if C_{Q}(y) ∈ Syl_{p}(N_{G}(Q) ∩ C_{G}(y)). ii) Suppose that p is odd, and let Q be a p-subgroup of G and y be an involution of N_{G}(Q). Then Q is a maximal y-invariant p-subgroup of G if and only if yQ neither inverts nor centralizes any element of order p in N_{G}(Q)/Q. iii) Suppose that p = 2, and let Q be a 2-subgroup of G and y be an element of order 3 in N_{G}(Q). Then Q is a maximal y-invariant 2-subgroup of G if and only if yQ is not contained in any subgroup isomorphic to A_{4} of N_{G}(Q)/Q, and yQ does not centralize any involution of N_{G}(Q)/Q.

Proof: i) Notice that Q is a maximal y-invariant p-subgroup of G if and only if Q is a maximal y-invariant p-subgroup of N_{G}(Q), for if Q < R and R is another y-invariant p-subgroup of G, then Q < N_{R}(Q) and N_{R}(Q) is y-invariant. Hence we may suppose that Q⊲G, and do so. Set Ḡ = G/Q, etc. Then \overline{C_{G}(y)} = CḠ(ȳ) since y is p-regular and Q is a p-group. Since ȳ ∈ O_{p′}(Ḡ), we see that ȳ centralizes any p-subgroup of Ḡ which it normalizes. Hence Q is a maximal yinvariant p-subgroup of G if and only if ȳ normalizes no non-trivial p-subgroup of Ḡ, if and only if ȳ centralizes no non-trivial p-subgroup of Ḡ, if and only if C_{Q}(y) ∈ Syl_{p}(C_{G}(y)).

ii) Again we may suppose that Q ⊲ G and we set Ḡ = G/Q. If ȳ inverts or centralizes an element of order p in Ḡ, then Q is clearly not a maximal yinvariant p-subgroup of G. On the other hand, if ȳ normalizes a non-trivial p-subgroup of Ḡ, then ȳ normalizes a non-trivial Abelian p-subgroup Ā, say. We have Ā = [Ā, ȳ] × CĀ(ȳ), so that ȳ must either centralize or invert a nonidentity element of Ā.

iii) The proof of this part is analogous to part ii), except that in the final step, Ā may be chosen to be elementary Abelian, and [Ā, ȳ] is a direct product of ȳ-invariant Klein 4-groups, each acted on by ȳ without non-trivial fixed points.

Definition: When p is an prime and G is a finite group, a pair (Q, x) is called a pseudoweight for G if Q is a maximal x-invariant p-subgroup of G, and one or more of the following occurs:

i) xQ ∈ O_{p′}(N_{G}(Q)/Q). ii) p is odd and x is an involution. iii) p = 2 and x has order 3.

Remark: It is easy to check that (Q, 1) is a pseudoweight for G if and only if Q ∈ Syl_{p}(G), so there is a unique conjugacy class of pseudoweights with second component 1_{G}. When Q is a Sylow p-subgroup of G, notice that the number of non-conjugate pseudoweights with first component Q is the number of conjugacy classes of p-regular elements of N_{G}(Q), since N_{G}(Q)/Q is a p′-group. If p is odd, every involution occurs as the second component of at least one pseudoweight, since whenever t is an involution, there is at least one maximal t-invariant psubgroup of G (which may be trivial). If p = 2, then every element of order 3 occurs as the second component of at least one pseudoweight, since whenever u ∈ G has order 3, there is at least one maximal u-invariant 2-subgroup of G.

Before our first result, we recall some results of (Murray,[5]) and [6]. Let P be a Sylow p-subgroup of G. In [6], it is proved that the number of p-blocks of defect zero is the rank of a matrix S with entries in GF(p) defined as follows: the rows and columns of S are indexed by the conjugacy classes of p-regular elements y of G such that C_{G}(y) is a p′-group. The (i, j)-entry of S is s_{ij}, which is the residue (mod p) of \frac{|Ω_{ij}|}{|P |}, where Ω_{ij} = \{(u, v) ∈ C_{i} × C_{j} such that u^{−1}v ∈ P }, C_{i} being the i-th conjugacy class of p-regular elements of p-defect zero. This is refined by 6.3 of (Murray,[5]): Ω_{ij} may be replaced by Ω̃_{ij}, which is obtained by only counting ordered pairs (u, v) such that u^{−1}v is an element of P of order at most p, and we may use S̃ in place of S, where s̃_{ij} is the residue (mod p) of \frac{|Ω̃_{ij}|}{|P |}. We will see that, when p = 2, this refinement is advantageous.

Theorem 1: For a given prime p, the number of non-conjugate weights of G is greater than or equal to the number of non-conjugate pseudoweights of G.

Proof: First note that (for any prime p), G permutes its pseudoweights by conjugation. Let Q be a p-subgroup of G. Then the G-conjugate pseudoweights with first component Q correspond bijectively to the N_{G}(Q)/Q-conjugacy classes of pseudoweights with first component the trivial group. For there is a bijection between p-regular conjugacy classes of N = N_{G}(Q) and p-regular conjugacy classes of N̄ = N/Q, and we have \overline{C_{N}(x)} = CN̄(x̄) for each p-regular element x of N. Hence we are reduced to proving that the number of p-blocks of defect zero is at least the number of conjugacy classes of pseudoweights with first component trivial.

Let (1, x_{1}), …, (1, x_{d}) be representative for the conjugacy classes of pseudoweights of G with trivial first component. Then no x_{i} normalizes any nontrivial p-subgroup of G.

Let us label so that x_{i} ∈ C_{i} for 1 ≤ i ≤ d. We show that the first d × d minor of S̃ is an invertible diagonal matrix, so that S̃ has rank at least d. For if 1 ≤ i, j ≤ d and u is conjugate to x_{i}, v is conjugate to x_{j} with u^{−1}v ∈ P of order at most p, then u^{−1}v is p-regular (if u or v is in O_{p′}(G) this is clear. If p is odd and u and v are both involutions which invert no element of order p, then u^{−1}v must be p-regular. If p = 2 and u and v are both elements of order 3 which normalize no non-trivial 2-subgroup of G and u^{−1}v is an involution then ⟨u, v⟩ ≅ A_{4} and u is conjugate to v within ⟨u, v⟩, a contradiction). Hence u^{−1}v is p-regular in all cases, (so is the identity, as P is a p-group). Thus s̃_{ij} = 0 for i ≠ j and 1 ≤ i, j ≤ d. Also, s̃_{ii} is the residue (mod p) of \frac{|C_{i}|}{|P |} for 1 ≤ i ≤ d. Thus s̃_{ii} ≠ 0 for 1 ≤ i ≤ d, as required to complete the proof.

Because of its analogy with the result of Brauer-Fowler [1] mentioned previously, we single out for special mention:

Corollary 1: Let G be a finite group of order divisible by 6. If G contains an element of order 3 which normalizes no non-trivial 2-subgroup of G, then G has a 2-block of defect zero. More precisely, the number of 2-blocks of defect zero is greater than or equal to the number of conjugacy class of elements of order 3 of G which normalize no non-trivial 2-subgroup of G.

We now combine some local-group theoretic analysis with the block-theoretic results we have used.

Theorem 2: Let G be a finite group of even order which contains no element of order 2p for some odd prime p. Then either G has a p-block of defect zero or else G has Abelian Sylow p-subgroups and a unique conjugacy class of involutions. Furthermore, if G has no p-block of defect zero, and has Sylow p-subgroups of rank at least 3, then either G/O_{{2,p}′}(G) has a normal Sylow p-subgroup or else G has a strongly p-embedded subgroup.

Proof: Suppose that G has no p-block of defect zero. Set π = {2, p}. To prove the Theorem, it suffices to consider the case that O_{π′}(G) = 1. By the result of Brauer-Fowler mentioned earlier, every involution of G inverts an element of order p, as G has no element of order 2p. Also, since G contains no element of order 2p, no section of G is isomorphic to SL(2, p), so that (by a Theorem of Glauberman,[3]) N = N_{G}(ZJ(P)) controls strong fusion in G for P ∈ Syl_{p}(G). Thus N must have even order, as some element of order p is conjugate to its inverse in G.

Since G contains no element of order 2p, the Sylow 2-subgroups of N must be cyclic or generalized quaternion by the Brauer-Wielandt formula. Hence N has a unique conjugacy class of involutions and (by the Brauer-Suzuki Theorem) N = O_{2′}(N)C_{N}(t) for t any involution of N. Thus O_{2′}(N) contains P as C_{N}(t) is a p′-group. We may suppose that P is t-invariant, so that P is Abelian as t acts without non-trivial fixed-points on P. We wish to prove that G has a unique conjugacy class of involutions. Let u be an involution of G. Then, replacing u by a conjugate if necessary, we may suppose that u inverts an element h of order p in P. Then N_{G}(⟨h⟩) = C_{G}(h)N_{N}(⟨h⟩) so that u is conjugate within N_{G}(⟨h⟩) to an involution of N_{N}(⟨h⟩) as C_{G}(h) has odd order. In particular, u is conjugate in G to an involution of N. This completes the proof of the first claim, as N has one conjugacy class of involutions.

For the second claim, set A = Ω_{1}(P), and suppose that |A| ≥ p^{3}. For each a ∈ A^{#}, we know that C_{G}(a) has odd order by hypothesis, so is solvable. Thus C_{G}(a) = C_{N}(a)O_{p′}(C_{G}(a) for each such a. If O_{p′}(C_{G}(a)) = 1 for each such a, then either N is strongly p-embedded in G or else P ⊲ G ( for if O_{p}(G) ≠ 1, then G = O_{2′}(G)C_{G}(t) for t an involution, and O_{2′}(G) has a normal Sylow psubgroup as O_{π′}(G) = 1). Otherwise, by the solvable signalizer functor theorem (Glauberman,[2]),

θ(A) = ⟨O_{p′}(C_{G}(a)) : a ∈ A^{#}⟩

is a solvable π′-group. Then M = N_{G}(θ(A)) < G. Now N = N_{G}(P) ≤ N_{G}(A) ≤ M. Also, for each a ∈ A^{#}, we have C_{G}(a) = C_{N}(a)O_{p′}(C_{G}(a)) ≤ M. For each non-trivial subgroup B of P, we have

N_{G}(B) ≤ N_{G}(Ω_{1}(B)) = C_{G}(Ω_{1}(B))N_{N}(Ω_{1}(B)) ≤ M.

Thus M is strongly p-embedded in this case.

REFERENCES

1. R. Brauer and K. A. Fowler, On groups of even order, Ann. of Math. 62 (1955), 565-583.

2. G Glauberman, On solvable signalizer functors in finite groups, Proc. London Math. Soc. (3) 33 (1976), no. 1, 1-27.

3. G. Glauberman, A characteristic subgroup of a p-stable group, Canad. J. Math. 20 1968 1101-1135.

4. J. Murray, Projective modules and involutions, J. Algebra 299 (2006), no. 2, 616-622.

5. J. Murray, Blocks of Defect Zero and Products of Elements of Order p, Journal of Algebra 214 (1999), no. 2, 385-399.

6. G.R. Robinson, The number of blocks with a given defect group, J. Algebra 84 (1983), no. 2, 493–502.

7. Y. Tsushima, On the weakly regular p-blocks with respect to O_{p′}(G), Osaka J. Math. 14 (1977), 465-470.

8. T. Wada, On the existence of p-blocks with given defect groups, Hokkaido Math. J. 6 (1977), 243-248.

Appendix

Corollary 2: Let G be a {2, 3}-group whose Sylow 3-subgroups have exponent 3. Then for each 3-element x ∈ G, the maximal x-invariant 2-subgroups of G are transitively permuted under conjugation by C_{G}(x).

Proof: Suppose that p = 2. Then Alperin’s weight conjecture holds for p and G, as G is solvable. Hence the number of (conjugacy classes of) weights for G is the number of conjugacy classes of 3-elements of G. On the other hand, the number of (conjugacy classes of) pseudoweights for G is the number of conjugacy classes of pairs (Q, x) with x an element of order at most 3 and Q a maximal x-invariant 2-subgroup of G, and this is less than or equal to the number of (conjugacy classes of) weights. On the other hand, this number is greater than or equal to the number of conjugacy classes of 3-elements, which is equal to the number of weights of G (for p = 2). Hence the number of (conjugacy classes of) pseudoweights for G is equal to the number of conjugacy classes of 3-elements of G, so for each 3-element x of G, there is exactly one C_{G}(x)-orbit by conjugation on pairs (Q, x) such that Q is a maximal x-invariant 2-subgroup of G.