A group theoretical characterization of simple, locally finite, finitary linear groups
As the title implies, the aim of the paper is to obtain a straightforward group theoretical property that singles out the finitary linear groups from among the simple locally finite groups. A finitary linear group is by definition a group that can operate faithfully on a vector space in such a way that each element operates trivially on a subspace of finite codimension. The property obtained is as follows. Theorem 1. Let \(G\) be a simple non-abelian locally finite group. Then \(G\) is finitary linear if and only if there exists an involution \(x \in G\) and an integer \(d \geq 1\) such that if \(S\) is any 2-subgroup \(G\) containing \(x\), then the normal closure of \(x\) in \(S\) is soluble of derived length at most \(d\). The fact that this condition holds if \(G\) is finitary linear is a recent result of \textit{U. Meierfrankenfeld, R. E. Phillips} and \textit{O. Puglisi} [J. Lond. Math. Soc., II. Ser. 47, 31-40 (1993; Zbl 0738.20043)]. The converse depends on the classification of finite simple groups. Theorem 1 breaks down as it stands if 2 is replaced by an odd prime. In fact, for any odd prime \(p\), there is a simple locally finite group that is not finitary linear, but nevertheless contains an element \(x\) of order \(p\) whose normal closure, in any \(p\)-subgroup containing it, is abelian. Two questions about analogous characterizations of simple locally finite linear groups are raised in the paper. They have been answered affirmatively in a remarkable preprint of \textit{U. Meierfrankenfeld} [Michigan State Univ.] which contains other very interesting results about simple locally finite groups.
scientific article; zbMATH DE number 571904
scientific article