On the Centralizer of an Element of Order Four in a Locally Finite Group

Abstract

Abstract. We prove that if G is a locally finite group admitting an automorphism φ of order four such that C G (φ) is Chernikov, then G has a soluble subgroup of finite index.

Locally finite groups containing a $$2$$ 2 -element with Chernikov centralizer

“…In view of the difficulty of the above problems, it makes sense to study them under certain additional conditions. In the present paper we extend the result of [21] for elements of order 2 n . Our main result is the following theorem.…”

“…Another problem is whether one can replace "locally soluble" by "soluble" in such results. This problem was solved in the positive for an element of prime order and order 4 in [20] and [21], respectively.…”

“…Theorem 1.2 in turn is a strengthening of Theorem 1.10 in [22], which under the same hypotheses was giving a soluble subgroup of bounded index whose derived length was bounded in terms of m, n, c. It is elimination of the dependence of the derived length on m in Theorem 1.2 that made possible its application in the proof of Theorem 1.1. Similarly, the aforementioned theorems in [20,21] on locally finite groups containing an element of prime order p or of order 4 with Chernikov centralizer used the results with 'strong' bounds on finite groups containing an element of order p or 4 with finite centralizer. Namely, in the case of finite centralizer such a group contains a subgroup of bounded index that is nilpotent (or soluble) of nilpotency class (or derived length) bounded in terms of the order of the element only.…”