# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 132, 2014, pp. 33–43**

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

Harmonic numbers and finite groups

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

^{[2]}(1) Department of Mathematics, The North Eastern Hill University Library, Permanent Campus, Shillong, 793022, Meghalaya, India

(2) Department of Mathematics, The North Eastern Hill University Library, Permanent Campus, Shillong, 793022, 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