Non-abelian Sylow subgroups of finite groups of even order

Abstract

We have been able to prove that every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. The proof depends upon the consequences of the classification of finite simple groups. The purpose of this note is to announce [4]: Main Theorem. Every nonabelian Sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution. Let G be a finite group and Γ(G) the prime graph of G. Γ(G) is the graph such that the vertex set is the set of prime divisors of |G|, and two distinct vertices p and r are joined by an edge if and only if there exists an element of order pr in G. Let n(Γ(G)) be the number of connected components of Γ(G) and dG(p, r) the distance between two vertices p and r of Γ(G). It has been proved that n(Γ(G)) ≤ 6 in [12], [10], [11], [9]. Theorem 1. Let G be a finite group of even order and p a prime divisor of |G|. If dG(2, p) ≥ 2, then a Sylow p-subgroup of G is abelian. Theorem 1 is a restatement of the Main Theorem in terms of the prime graph Γ(G). Corollary 1. Let G be a finite group of even order and p a prime divisor of |G|. If ∆ is a connected component of Γ(G)−{p} not containing 2, then a Sylow r-subgroup of G is abelian for any r ∈ ∆. There is a certain relation between a subgraph Γ(G) − {p} of Γ(G) and Brauer characters of p-modular representations of G (see [3]). Theorem 2. Let G be a finite nonabelian simple group and p an odd prime divisor of |G|. Then dG(2, p) = 1 or 2 provided dG(2, p) < ∞. The significance of the prime graphs of finite groups can be found in [1], [3], [5], [6], [7], [8], [14], [15]. We apply the classification of finite simple groups (see [1], [2], [4], [7], [10], [13]). It has been proved that a minimal counterexample to Theorem 1 Received by the editors October 20, 1997. 1991 Mathematics Subject Classification. Primary 20D05, 20D06, 20D20.