A general Brauer-Fowler theorem and centralizers in locally finite groups
A classical theorem of Brauer and Fowler, proved by quite elementary methods, states that the order of a finite non-abelian simple group G is bounded in terms of the order of the centralizer of any involution in G. Here we use the classification of finite simple groups to verify that the order of such a G is bounded in terms of the order of any automorphism \(\alpha\) of G and the number of fixed points of \(\alpha\). It follows easily that if a locally finite group G contains an element x with finite centralizer, then it has a locally soluble subgroup H of finite index. In particular, if G itself is simple, then it is finite. If, further, x has prime power order \(p^ n\), then we may take \(H=O_{p'p}(H)=F_ n(H)\), where \((F_ i(H))\) is the Hirsch-Plotkin series of H. As a further application of the techniques used to prove the above, we improve a result of the author [J. Lond. Math. Soc., II. Ser. 37, 421-436 (1988; Zbl 0619.20018)]. We show that if G is an infinite simple group of Lie type over a locally finite field K of characteristic p, and \(\alpha\) is an automorphism of G of finite order not divisible by p, then there exist infinitely many primes q such that \(\alpha\) fixes an element of order q of G. This was proved previously under restrictions on the order of \(\alpha\).
order of a finite non-abelian simple group
locally soluble subgroup H of finite index
scientific article; zbMATH DE number 4173236