Harmonic numbers and finite groups

Abstract

Given a finite group $G,$ let $\tau (G)$ be the number of normal subgroups of $G$ and $\sigma (G)$ be the sum of the orders of the normal subgroups of $G$ . The group $G$ is said to be harmonic if $H(G):=|G|\tau (G)/\sigma (G)$ is an integer. In this paper, all finite groups for which $1\le H(G)\le 2$ have been characterized. Harmonic groups of order $pq$ and of order $pqr$ , where $p<q<r$ are primes, are also classified. Moreover, it has been shown that if $G$ is harmonic and $G\ncong C_6$, then $\tau (G)\ge 6$.