Sum of Element Orders on Finite Groups of the Same Order

Abstract

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group Cn. In this paper, we investigate the minimum value of ψ on the set of groups of the same order.

Sums of prime element orders in finite groups

“…Proof: The proof of this theorem is similar to that of Amiri and Jafarian Amiri mentioned in [1,Theorem]. But we prefer to write it step by step to clarify certain changes for the reader.…”

“…Proof: The equivalence (1)⇐⇒(2) is a direct consequence of the previous theorem, and the equivalence In this section, we investigate the minimum and the maximum value of c * on the set of groups of the same order. We use the results of the previous section and the properties of cyclic groups to prove the first main theorem (see Theorem 3.3), and to prove the second main theorem (see Theorem 3.7), we use some ideas inspired by the work of Amiri and Jafarian Amiri [1].…”

“…Motivated by the works of Amiri, Jafarian Amiri and Isaacs [1][2][3][4] and Shen et al [5] in the study of c(G)the sum of element orders of a finite group G, we will introduce, in this paper, another function denoted by c * (G), which is the sum of prime element orders. More precisely, the function c * is defined as follows:…”

Sums of Element Orders in Symmetric Groups

“…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.…”

“…Our main starting point is given by the papers H. Amiri et al (2009), H. Amiri and S.M.J. Amiri (2011) which studied on the sums of element orders in finite groups. Given a finite group , we denote the sum of element orders in by ( ).…”