# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 132, 2014, pp. 33–43**

**DOI: 10.4171/RSMUP/132-3**

Published online: 2014-11-04

Harmonic numbers and finite groups

Sekhar Jyoti Baishya^{[1]}and Ashish Kumar Das

^{[2]}(1) The North Eastern Hill University Library, Meghalaya, India

(2) The North Eastern Hill University Library, Meghalaya, India

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):=\vert G\vert {\tau} (G)/{\sigma} (G)$ is an integer. In this paper, all finite groups for which $ 1 \leq H(G) \leq 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 \not \cong C_ 6$, then $ {\tau} (G) \geq 6$.

*Keywords: *Finite groups, harmonic numbers, harmonic groups

Baishya Sekhar Jyoti, Das Ashish Kumar: Harmonic numbers and finite groups. *Rend. Sem. Mat. Univ. Padova* 132 (2014), 33-43. doi: 10.4171/RSMUP/132-3