On the Existence of $f$-local Subgroups in a Group with Finite Involution

Abstract

An $f$-local subgroup of an infinite group is each its infinite subgroup with a nontrivial locally finite radical. An involution is said to be finite in a group if it generates a finite subgroup with each conjugate involution. An involution is called isolated if it does not commute with any conjugate involution. We study the group $G$ with a finite non-isolated involution $i$, which includes infinitely many elements of finite order. It is proved that $G$ has an $f$-local subgroup containing with $i$ infinitely many elements of finite order. The proof essentially uses the notion of a commuting graph.

Keywords

• group
• $f$-local subgroup
• finite involution
• commuting graph