Turkish Journal of Mathematics

Abstract: The topic of the present paper is the following question. Let G be a locally finite group admitting an automorphism f of finite order such that the centralizer C_G(f) satisfies certain finiteness conditions. What impact does this have on the structure of the group G? Equivalently, one can ask the same question when f is an element of G. Sometimes the impact is quite strong and the paper is a survey of results illustrating this phenomenon. In particular, we concentrate on results where G is shown to have a large nilpotent or soluble subgroup. Naturally, in each case the result depends on the order of the automorphism f and kind of conditions imposed on C_G(f). We shall be considering mostly the classical finiteness conditions such as C_G(f) being finite, Chernikov, and of finite rank, respectively. It is not a purpose of the paper to survey numerous results on automorphisms of finite groups. In particular, among important topics that are left out of the present discussion are ``p-automorphisms of p-groups\" (see [36]) and ``length problems\" (see [73]). However, in some situations (like, for example, when C_G(f) is finite) problems on infinite groups quickly reduce to finite groups and in those cases working with finite groups is very natural. A separate section of the paper is devoted to the case when f is of order two, a prime, four, or other, respectively. However, before anything else we address in Section 1 the following related question. Given a periodic group G with an automorphism f, what additional assumptions on G and C_G(f) ensure that G is locally finite?

Key Words: Locally finite groups, centralizer