Variants of some of the Brauer-Fowler theorems

Brauer and Fowler noted restrictions on the structure of a finite group Gin terms of vertical bar C-G(t)vertical bar for an involution t is an element of G. We consider variants of these themes. We first note that for an arbitrary finite group Gof even order, we have vertical bar G vertical bar < k(F)vertical bar C-G(t)vertical bar(4) for each involution t is an element of G, where Fdenotes the Fitting subgroup of Gand k(F) denotes the number of conjugacy classes of F. In particular, for such a group Gwe have [G : F(G)] < vertical bar C-G(t)vertical bar(4) for each involution t is an element of G. This result requires the classification of the finite simple groups. The groups SL(2, 2(n)) illustrate that the above exponent 4 cannot be replaced by any exponent less than 3. We do not know at present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices when G is almost simple. We are however able to prove that every finite group G of even order contains an involution u with [G : F(G)] < vertical bar C-G(u)vertical bar(3). The proof of this fact reduces to proving two residual cases: one in which G is almost simple (where the classification of the finite simple groups is needed) and one when G has a Sylow 2-subgroup of order 2. For the latter result, the classification of finite simple groups is not needed (though the Feit-Thompson odd order theorem is). We also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups, and some other results on the existence of non-central elements (not necessarily involutions) with large centralizers in general finite groups. Lastly we prove (without using the classification of finite simple groups) that if G is a finite group and t is an element of G is an involution, then all prime divisors of [G : F(G)] are less than or equal to vertical bar C-G(t)vertical bar + 1. (C) 2019 Elsevier Inc. All rights reserved.