The structure of infinite 2-groups with a unique 2-element subgroup

TL;DR

This paper classifies infinite 2-groups with a unique 2-element subgroup, showing they are either quasicyclic 2-groups or infinite generalized quaternion groups.

Contribution

It provides a complete classification of infinite 2-groups with a single 2-element subgroup, identifying their isomorphism types.

Findings

Infinite 2-groups with a unique 2-element subgroup are either quasicyclic 2-groups or infinite generalized quaternion groups.

The classification is complete and definitive for this class of groups.

Abstract

We prove that each infinite 2-group with a unique 2-element subgroup is isomorphic either to the quasicyclic 2-group or to the infinite group of generalized quaternions.