Characterization of $p$-groups by sum of the element orders

Abstract

Let G be a finite group. Then we denote ψ(G) = ∑ x∈G o(x) where o(x) is the order of the element x in G. In this paper we characterize some finite p-groups (p a prime) by ψ and their orders.Theorem 1.1. Suppose that P and Q are contained in CP 2 of the same order p n . Then the following statements are equivalent:(1) ψ(P ) = ψ(Q).(2(3) ψ(Ω i (P )) = ψ(Ω i (Q)) for all i ∈ N.

The second maximal groups with respect to the sum of element orders

“…Notice that if q = 2, then f (q) = 21 33 = 7 11 . Moreover, if q is an odd prime, then q > 2 and by our previous remark f (q) < f (2) = 7 11 . Hence it follows by Theorem 4 that if G is a q * -group for some odd prime q, then…”

“…Hence q = 2, p = 3 and in particular n2 )( n 3 ). But this inequality implies, as shown in the proof of Theorem 1 in [12], that ψ(G) < 7 11 ψ(C n ) = f (2)ψ(C n ), in contradiction to our assumptions.…”

Another criterion for solvability of finite groups

“…Then o(G) ≥ 3.55. q+1 and i 2 (G) ≤ 2(q + 1)q 13 . Since q ≥ 2, we have i 2 (G) ≤ 2(q + 1)q 13 ≤ 2(q + 1)q 24 (q 2 − 1)(q 2 − 1)q 7 = 2q 24 (q − 1) 2 (q + 1)q 7 ≤ 2q 24 100(q + 1)…”

On the supersolvability of a finite group by the sum of subgroup orders

“…x∈G o(x) (where o(x) denotes the order of the element x), have been recently investigated by many authors (see e.g. [1,2,3]).…”

“…. , m, are finite groups of coprime orders, then σ 1 ( m i=1 G i ) = m i=1 σ 1 (G i ); -σ 1 (G) ≥ σ 1 (G/H) + 1 (G:H) (σ 1 (H) − 1) ≥ σ 1 (G/H), for all H G.…”