Hamiltonian Groups with Perfect Order Classes

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 $k$.

9 pages, no figures; Feedback is welcome v2 Submitted version (replaced one lemma with citation to previous occurrence in the literature; other textual emendations) v3 8 pages, no figures: Simplified proof of one lemma and other minor edits