Harmonic numbers and finite groups

Abstract

Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of the orders of the normal subgroups of G. The group G is said to be harmonic if H(G) : jGjt(G)=s(G) is an integer. In this paper, all finite groups for which 1 H(G) 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 T C 6 , then t(G) ! 6.