Simple locally finite groups with the minimal condition for 2-subgroups (Q790249)

The following theorem is proved. Let G be an infinite simple locally finite group satisfying the minimal condition on 2-subgroups and such that \(C_ G(i)\) is almost locally soluble for every involution \(i\in G\). Then \(G\cong PSL(2,F)\) for some infinite locally finite field F of odd characteristic. This answers a question raised by Kegel in ''Kourovka Notebook'' [see Zbl 0509.20001 for a review of the 8th edition (1982)]. The same conclusion has recently been obtained by V. Turau (as yet unpublished) with the same hypotheses except that \(C_ G(i)\) is assumed almost locally soluble only for at least one involution \(i\in G\) rather than all involutions. The theorem is proved by showing that the centralizer of an arbitrary involution in G satisfies Min-p for all primes p, then a result of \textit{V. V. Belyaev} [Algebra Logika 20, 605-619 (1981) \(=\) Algebra Logic 20, 393-402 (1981; Zbl 0489.20032)] gives the theorem immediately. The statement about centralizers of involutions follows by a simple argument from results on the B-conjecture and facts about known finite simple groups.