Finite groups with small unbalancing 2-components

Robert Hugh Gilman and Ronald Mark Solomon

Vol. 83 (1979), No. 1, 55–106

DOI: 10.2140/pjm.1979.83.55

Abstract

In this paper we supply one link in a chain of results which will prove the following two conjectures:

B(G)-Conjecture. If H is a 2-local subgroup of a finite group G, then [L(H),O(H)] ⊆ O(G).

Unbalanced Group Conjecture. If G is a finite group with O(C_G(t))⊈O(G) for some involution t ∈ G, then O(C_G(t)) acts nontrivially on L∕Z^∗(L) where L is a 2-component of G with L∕Z^∗(L) isomorphic to one of the following simple groups:

1. A simple Chevalley group or twisted variation over a field of odd order;
2. An alternating group of odd degree;
3. PSL (3,4) of He, the simple group of Held.

Mathematical Subject Classification 2000

Primary: 20D05

Milestones

Received: 25 July 1977

Revised: 2 October 1978

Published: 1 July 1979