Sums of element orders in groups of order 2m with m odd

Abstract

If G is a finite group, then ψ(G) denotes the sum of orders of all elements of G and if k is a positive integer, then C_k denotes a cyclic group of order k. Moreover, ψ(C_k) will be sometimes denoted by ψ(k). In this article we deal with groups of order $n=2m$ with m odd. Our main results are the following two theorems: Theorem 7. Let G be a non-cyclic group of order $n=2m$, with m an odd integer. Then $\psi (G)\le \frac{13}{21}\psi (C_n)$. Moreover, $\psi (G)=\frac{13}{21}\psi (C_n)$ if and only if $G=S_3\times C_{n/6}$, where $n=6m_1$ with $(m_1,6)=1$ and S₃ is the symmetric group on three letters. Theorem 8. Let Δ_n be the set of non-cyclic groups of the fixed order $n=2m$, where m is an odd integer, and suppose that $m=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_t^{{\alpha }_t}$, where p_i are distinct primes and α_i are positive integers for all i. If $G\in {\Delta }_n$, then $\psi (G)\le (\frac{1}{3}+\frac{2l}{3\psi (l)})\psi (C_n)$, where $l=min\left\{p_i^{{\alpha }_i}\right|i\in \{1,\dots ,t\}\}$. Moreover, $G\in {\Delta }_n$ satisfies $\psi (G)=(\frac{1}{3}+\frac{2l}{3\psi (l)})\psi (C_n)$ if and only if $G=D_{2l}\times C_{n/2l}$, where $D_{2l}$ is the dihedral group of order 2l.

Key words:

• Group element orders
• groups of even order

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60
• 20E34

Additional information

Funding

This work was supported by the National Group for Algebraic and Geometric Structures, and their Applications (GNSAGA - INDAM), Italy. The first author is grateful to the Department of Mathematics of the University of Salerno for its hospitality and support, while this investigation was carried out.