On locally soluble periodic groups with Chernikov centralizer of a four-subgroup

Abstract

Let G be a locally soluble periodic group having a four-subgroup V. We show that if CG(V) is Chernikov then G is hyperabelian-by-Chernikov, if CG(V) is finite then G is hyperabelian.

On locally finite groups with a small centralizer of a four-subgroup

“…So the equality [G, v 3 ] = J 1 , J 2 follows from Lemma 2.3 (4). The fact that [G, V ] = J 1 , J 2 , J 3 is immediate from Lemma 5 of [20].…”

“…Proof. We remark that Lemma 4 from [20] shows that J 1 , J 2 is precisely the subgroup generated by all elements of G that v 3 takes to their inverses. So the equality [G, v 3 ] = J 1 , J 2 follows from Lemma 2.3 (4).…”

“…Recall that a group G is said to be hyperabelian if it has an ascending invariant series with abelian quotients. It was proved in [20] that a periodic locally soluble group containing a four-subgroup with finite centralizer is hyperabelian. Combining this with our results we obtain the following corollary.…”