Exact bounds for the sum of the inverse-power of element orders in non-cyclic finite groups

2024/04/11 by Archita, M.
#20 Group theory and generalizations #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2404.07500

Abstract

Given a finite group G of order n. Denote the sum of the inverse-power of element orders in G by m(G). Let ℤ_n be the cyclic group of order n. Suppose G is a non-cyclic group of order n then we show that m(G)≥ (5)/(4)m(ℤ_n). Our result improves the inequality m(G)>m(ℤ_n) obtained by Baniasad Azad, M., and Khorsravi B. Moreover, this bound is best as for n=4l, l odd, there exists a group G of order n satisfying m(G)=(5)/(4)m(ℤ_n). Moreover, we will establish that (1)/(q-1)m(G)< m(ℤ_n)≤ (4)/(5)m(G), where G is a non-cyclic group of odd order.

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