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- W2947291051 abstract "Denote by $G$ a finite group and by $psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $psi(t)=psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $frac {psi(G)}{psi(|G|)}leq frac {85}{301}$, and equality holds if and only if $n=3cdot 7cdot m_1$ with $(m_1,42)=1$ and $G=(C_7rtimes C_3)times C_{m_1}$, with $C_7rtimes C_3$ non-abelian. (2) If $q>3$, then $frac {psi(G)}{psi(|G|)}leq frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_qtimes C_ptimes C_p times C_{m_1}$." @default.
- W2947291051 created "2019-06-07" @default.
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- W2947291051 date "2019-05-29" @default.
- W2947291051 modified "2023-09-27" @default.
- W2947291051 title "Sums of element orders in groups of odd order" @default.
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