A remark on {2,3}-groups with no elements of order six

TL;DR

This paper characterizes certain {2,3}-groups with restrictions on element orders and centralizers, proving they are locally finite, which advances understanding of their structure and properties.

Contribution

It provides a detailed description of {2,3}-groups with specific order and centralizer conditions, establishing their local finiteness for the first time.

Findings

Groups are locally finite under given conditions.

Product of elements of order at most 4 has order at most 9.

Centralizers of involutions are locally cyclic 2-subgroups.

Abstract

We describe $\{2,3\}$-groups in which the order of a product of any two elements of orders at most $4$ does not exceed $9$ and the centralizer of every involution is a locally cyclic $2$-subgroup. In particular, we will prove that these groups are locally finite.