An exact upper bound for sums of element orders in non-cyclic finite groups

作者：Herzog Marcel; Longobardi Patrizia^*; Maj Mercede

来源：Journal of Pure and Applied Algebra, 2018, 222(7): 1628-1642.

DOI：10.1016/j.jpaa.2017.07.015

Denote the sum of element orders in a finite group G by psi(G) and let C-n, denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that psi(G) <= 7/11 psi(C-n) and psi(G) < 1/q-1 psi(C-n). The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying psi(G) = 7/11(C-n) and the second result implies that if G is of odd order, then ING) < 1/2 psi(C-n). Our results improve the inequality psi(G) < psi(C-n) obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some tP(G)-based sufficient conditions for the solvability of G.

• 出版日期2018-7