Finite groups with many involutions

TL;DR

This paper characterizes finite groups with a high proportion of involutions, showing that if more than 75% are involutions, the group must be elementary abelian 2-group, and describes the structure when exactly 75% are involutions.

Contribution

It provides a complete characterization of finite groups based on the proportion of involutions, identifying the structure of groups with exactly 75% involutions.

Findings

Groups with >75% involutions are elementary abelian 2-groups.

Groups with exactly 75% involutions are direct products of dihedral group of order 8 and elementary abelian 2-group.

The proportion of involutions determines the group's structure.

Abstract

It is shown that a finite group in which more than 3/4 of the elements are involutions must be an elementary abelian 2-group. A group in which exactly 3/4 of the elements are involutions is characterized as the direct product of the dihedral group of order 8 with an elementary abelian 2-group.