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

Abstract

Denote the sum of element orders in a finite group G by ψ(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 ψ(G) ≤ 7 11 ψ(C n ) and ψ(G) < 1 q−1 ψ(C n ). The first result is best possible, since for each n = 4k, k odd, there exists a group G of order n satisfying ψ(G) = 7 11 ψ(C n ) and the second result implies that if G is of odd order, then ψ(G) < 1 2 ψ(C n ). Our results improve the inequality ψ(G) < ψ(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 ψ(G)-based sufficient conditions for the solvability of G. the elements of G, there are many new and old results as well as many open 5 questions concerning ω(G) (see for example [9]). In [1] H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs introduced the function ψ(G), which denotes the sum of element orders of a finite group G, and proved that if G is a non-cyclic group of order n then ψ(G) < ψ(C n ), where C n denote the cyclic group of order n. Recently S.M. Jafarian Amiri and M. Amiri in [5] (see also [2] and [4]) and R.

A criterion for solvability of a finite group by the sum of element orders

“…Since the Shur multiplier of A 5 is 2 and G ′ is perfect, we get that either G ′ ∼ = A 5 or G ′ ∼ = SL (2,5). Now, we consider two cases:…”

Sums of Element Orders in Symmetric Groups

“…This lemma will be an important ingredient in the proof of our main result. For more details we refer to (Gorenstein 1968, Herstein 1958, Herzog et al 2018.…”

“…Sums of element orders in finite groups is an interesting subject, which was studied in varies papers (see Amiri (2009), Amiri and Amiri (2011), Herzog et al (2018)). Our main starting point is given by the papers H. Amiri et al (2009), H. Amiri and S.M.J.…”

Some Properties on Sums of Element Orders in Finite p-groups

“…Our main starting point is given by the papers (see H. Amiri [2], H. Amiri and S.M.J. Amiri [1], Herzog et al [5]) which studied on the sums of element orders in finite groups. Given a finite group G, we denote the sum of element orders in G by ψ(G).…”

“…Given a finite group G, we denote the sum of element orders in G by ψ(G). Historically, the most enlightening in this area is due [5], who introduced the function ψ(G) for a finite group G in [2] and proved that ψ(G) < ψ(C), where C denotes a cyclic group of the same order with the order of G. Then, in [5], by improving the results obtained by S.M. Jafarian Amiri and M. Amiri in [4] and by R. Shen, G. Chen and C. Wu in [10], M. Herzog, P. Longobardi and M. Maj found an exact upper bound for sums of element orders in non-cyclic finite groups.…”