Hamiltonian Groups with Perfect Order Classes

TL;DR

This paper characterizes finite Hamiltonian groups with perfect order classes, showing they are isomorphic to specific direct products involving quaternion and cyclic groups, providing a complete classification.

Contribution

It explicitly classifies all finite Hamiltonian groups with perfect order classes, identifying their precise algebraic structure.

Findings

Finite Hamiltonian groups with perfect order classes are isomorphic to Q×C_{3^k} or Q×C_2×C_{3^k}.

Such groups are characterized by their direct product structure involving quaternion and cyclic groups.

The classification is complete and explicit for all such groups.

Abstract

A finite group is said to have "perfect order classes" if the number of elements of any given order is either zero or a divisor of the order of the group. The purpose of this note is to describe explicitly the finite Hamiltonian groups with perfect order classes. We show that a finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic to the direct product of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a group of order at most $2$. Theorem. A finite Hamiltonian group has perfect order classes if, and only if, it is isomorphic either to $Q\times C_{3^k}$ or to $Q\times C_{2}\times C_{3^k}$, for some positive integer $k$.