Locally finite groups and their subgroups with small centralizers
A group \(G\) is almost locally soluble if \(G\) contains a locally soluble subgroup of finite index. \textit{B. Hartley} [J. Lond. Math. Soc., II. Ser. 37, No. 3, 421--436 (1988; Zbl 0619.20018)] proved that if a locally finite group contains an element of prime-power order with Chernikov centralizer, then it is almost locally soluble. The main result in this paper is Theorem 1.1: Let \(p\) be a prime and \(G\) a locally finite group containing an elementary abelian \(p\)-subgroup \(A\) of rank at least 3 such that \(C_{G}(A)\) is Chernikov and \(C_{G}(a)\) involves no infinite simple groups for any \(a \in A^{\#}\). Then, \(G\) is almost locally soluble. In view of the aforementioned result of Hartley [loc. cit.], the theorem remains valid also in the case where \(A\) is of prime order. On the other hand, Theorem 1.1 is no longer valid when \(A\) is of rank two. More precisely, the authors are able to prove Theorem 1.2: An infinite simple locally finite group \(G\) admits an elementary abelian \(p\)-group of automorphisms \(A\) such that \(C_{G}(A)\) is Chernikov and \(C_{G}(a)\) involves no infinite simple groups for any \(a\in A^{\#}\) if and only if \(G\) is isomorphic to \(\mathrm{PSL}_{p}(k)\) for some locally finite field \(k\) of characteristic different from \(p\) and \(A\) has order \(p^{2}\).
scientific article; zbMATH DE number 6714591
scientific article