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

In literature, there are many papers on the sum of element orders of a ﬁnite group. In this study, in particular, we deal with the cases in ﬁnite p -groups. Our main aim is to investigate the sums of element orders in ﬁnite p -groups and to give some properties of such sums. Let ψ ( G ) denote the sum of element orders of a ﬁnite group G . As an immediate consequence, we proved that ψ ( G ) < 3 4 ψ ( C ) and ψ ( G ) < 1 p − 1 ψ ( C ) , where G is a non-cyclic ﬁnite p -group of order p r and C is a cyclic group of order p r for some prime p


Introduction
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). 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. In [8], N. Mansuroglu derived the exact formula for the sums of element orders in symmetric groups. Throughout this paper, we assume that G is a finite p-group of order p r for a prime p. In this note we will focus on the study of ψ(G). Our main aim is to investigate the sum of element orders in finite p-groups and to give some properties of the sum of element orders in finite p-groups. We investigate to find an exact upper bound for sums of element orders in non-cyclic p-finite groups.

Preliminaries
This section contains necessary preliminary results and notation. We use standard notation. We define the function ψ(G) = x∈G o(x), where as usual, o(x) is the order of the element x. Basic concepts and some results on group theory can be found in [6,9]. Specially, more details for finite p-groups can be found in [7]. An important ingredient in our proofs is the following two lemmas which are particular case of Lemma 2.9 in the paper [5]. Lemma 2.1. If C is a cyclic group of order p r for some prime p, then Proof. Let ϕ(p) be Euler's function.

Lemma 2.2.
If C is a cyclic group of order p r for some prime p, then Proof. By Lemma 2.1,

The main results
Now, in this section we give our main results. Theorem 3.1. Let G be a non-cyclic finite p-group of order p r for some prime p. Then Proof. Since G is a non-cyclic finite p-group of order p r , for each element x ∈ G, o(x) ≤ p r−1 . But the order of the identity element 1 is 1, as a result of this, we have as required.

Theorem 3.2.
Let G be a non-cyclic finite p-group of order p r and C be a cyclic finite group of order p r . Then Proof. Suppose that ψ(G) ≥ 1 p−1 ψ(C). By Lemma 2.2, This implies that there exists x ∈ G with o(x) > p r−1 . Thus |G : x | < p and x is a p-group. As a consequence, |G| = o(x), namely G = x . But G is non-cyclic group, which is a contradiction. Hence, ψ(G) < 1 p−1 ψ(C).
We now provide a quantitative version of Theorem 3.2.
Corollary 3.1. Suppose that p is odd prime. Let G be a non-cyclic finite p-group of order p r and C be a cyclic finite group of order p r . Then ψ(G) < 1 2 ψ(C).
Some Properties on Sums of Element Orders in Finite p-groups

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The next result is the analogous of the result obtained by Theorem 1 in [5] for finite p-groups.

Theorem 3.3.
Let G be a non-cyclic finite p-group of order p r and C be a cyclic finite group of order p r . Then Proof. Assume that G is a non-cyclic finite p-group of order p r satisfying ψ(G) > 7 11 ψ(C). By Lemma 2.1, There exists x ∈ G with o(x) > 14 11(p+1) . It follows that |G : x | < 11(p + 1) 14 .
Now first we suppose that p = 2, then G is 2-group. Therefore, we have Thus |G : x | = 2, 2 r ≥ 4, r ≥ 2. Let C 2 r−1 be a cyclic group of order 2 r−1 . Hence which contradicts to the fact that ψ(G) > 7 11 ψ(C). Now we investigate the case that p ≥ 3. Since p is odd prime, it follows from Corollary 3.1 that which is a contradiction. This completes the proof.
Our main result is the following theorem. Proof. Assume that G is a non-cyclic finite p-group of order p r satisfying ψ(G) > 2 3 ψ(C). Since G is a non-cyclic, for each element x ∈ G, o(x) ≤ p r−1 . But the order of the identity element 1 is 1, as a result of this, we have ψ(G) ≤ (p r − 1)p r−1 + 1.
Now first we suppose that p = 2, then G is 2-group. Therefore, we have Now we investigate the case that p ≥ 3. Since p is odd prime, by Corollary 3.1, we have ψ(G) < 1 2 ψ(C) < 3 4 ψ(C), a contradiction. This completes the proof.
The following examples hold all our main consequences.  Then G is a generalized quarternion group of order 16. It is easy to see that G satisfies ψ(G) < 2 3 ψ(C) where C is a cyclic group of order 16.