Sums of prime element orders in finite groups

ABSTRACT Let G be a finite group and denote the sum of prime element orders of G. This paper presents some properties of and investigate the minimum value and the maximum value of on the set of groups of the same order.


Introduction
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: where o(x) is the order of the element x.
In Section 2, we give the preliminary definitions and results about the functions c(G) and c * (G). Particularly, we will show that if G 1 and G 2 are two finite groups, then c * (G 1 × G 2 ) = c * (G 1 ) + c * (G 2 ) if and only if the order of G 1 and that of G 2 are relatively prime, and as a consequence, we will prove that if G is a nilpotent group of order n, then c * (H) ≤ c * (G) for every nilpotent group H of order n if and only if every Sylow subgroup of G hasa prime exponent. Section 3 presents the main results of this work in the study of the minimum value and the maximum value of c * on the set of groups of the same order. More precisely, the main results are: (1) Let G be a finite group. Then c * (G) ≥ c * (C) for every cyclic group C of the same order as G.
(2) Let n be an integer which is not a nilpotent number and max{c * (G) | |G| = n} = c * (K) for some group K of order n. Then K is not nilpotent.

Preliminaries and basic results
This section presents some results and notations that will be useful in the sequel. Given a finite group G, let: . V(G) be the set of all positive divisor of |G|, . V * (G) be the set of all prime divisor of |G| and Definition 2.1: We define the area of G (the sum of element orders of G) as follows: Definition 2.2: We define the prime area of G (the sum of prime element orders of G) as follows: (1) c(G) = p( p + 1) + 1, (2) c * (G) = p( p + 1).
Proof: It is well known that if G is a nonabelian group of order 2p, where p is a prime number greater than or equal to 3, then G ≃ D 2p , and since the dihedral group D 2p has 1 element of order 1, p element of order 2, and p−1 element of order p, we obtain c(G) = p( p + 1) + 1 and c * (G) = p( p + 1). □ Lemma 2.6: Let G 1 and G 2 be two finite groups, then Proof: where e 1 and e 2 are, respectively, the identity element of G 1 and G 2 . Then, we have Hence □ Theorem 2.7: Let G 1 and G 2 be two finite groups. Then, the following statements are equivalent: It follows that Hence Reciprocally, assume that c * (G 1 × G 2 ) = c * (G 1 ) + c * (G 2 ). Using the previous lemma, we obtain Hence, for all prime number p we have |S p (G 1 )| = 0 or |S p (G 2 )| = 0. Consider the contrary that means(|G 1 |, |G 2 |) = 1. Then, there is a prime number p dividing both |G 1 | and |G 2 |. Applying the Cauchy theorem, there are elements x [ G 1 and y [ G 2 such that |x| = |y| = p. This is a contradiction to where p and q are distinct prime numbers, then: Proof: By the Cauchy theorem, there exists an element a (resp. b) in G of order p (resp. q). Let H = kal and K = kbl, then Since |H > K| = 1, we get HK=G. Then, the map f : H × K − G defined by f (x, y) = xy is an isomorphism. Applying Theorem 2.7, we obtain c * (G) = c * (H) + c * (K) = p( p − 1) + q(q − 1). (14) □ Theorem 2.9: Let G be a nilpotent group of order n. Then, the following are equivalent: (1) c * (H) ≤ c * (G) for every nilpotent group H of order n.
(2) Every Sylow subgroup of G has a prime exponent.
Proof: Put n = p n 1 1 · · · p n r r where p 1 , . . . , p r are distinct primes and n i are positive integers. Recall that a group is nilpotent if and only if it is the direct product of its Sylow subgroups [6,p.126]. Let H be a nilpotent group of order n. Then where P i is the Sylow p i -subgroup of H. From Theorem 2.7, we obtain Therefore If every Sylow subgroup of G has a prime exponent, then □ Corollary 2.10: Let G be a finite group of order n = p n 1 1 · · · p n r r where p 1 , . . . , p r are distinct primes and n i are positive integers. Then the following statements are equivalent: 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].
Lemma 3.1: Let p be a prime number that divides |G|. Then, |S p | is a multiple of ( p − 1).
Proof: Assume that |S p | = r and S p = {a 1 , · · · a r }. Let for all i = 1, . . . , r, T i = ka i l the cyclic subgroup of G generated by a i . Let T * i = T i \ {1}. Clearly every element in T * i is of order p, and for every i,j, Since the set T * i contain ( p − 1) elements of order p, we conclude that □ Corollary 3.2: Let G be a finite group of order n = p n 1 1 · · · p n r r , where p 1 , . . . , p r are distinct primes and n i are positive integers. Then there exist positive integers k i , . . . , k r such that Theorem 3.3: Let G be a finite group. Then c * (G) ≥ c * (C) for every cyclic group of the same order as G.
Proof: Let φ be Euler's phi-function. It is well known that if C is a cyclic group of order n, and r is a positive divisor of n, then the group C has w(r) elements of order r. Then Using Corollary 3.2, we obtain that c * (G) ≥ c * (C).
In the following, if d is a positive integer, we say that d satisfy the property N(d) if max{c * (G) | |G| = d} = c * (K) for some group K of order d, then K is not nilpotent. □ Lemma 3.4: Let d be a positive integer that satisfies the property N(d). Then n=ds satisfy the property N(n) for all positive integers s such that gcd(d, s) = 1.
Proof: Let G be a group of order n=ds such that c * (G) = max{c * (H) | |H| = n}.
If G is nilpotent, then G can be written as G = G 1 × G 2 , where |G 1 | = d and |G 2 | = s. By hypothesis, there exists a not nilpotent group K of order d such that c * (K) . c * (G 1 ). Let H = K × G 2 . Then H is a not nilpotent group of order ds and  Theorem 3.7: Let n be an integer which is not a nilpotent number. Assume that max{c * (G) | |G| = n} = c * (K) for some group K of order n. Then K is not nilpotent.
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. If n is not a nilpotent number, then there exists a group G not nilpotent of order n, two prime numbers p and q in V * (G), and an integer i such that p | q i − 1 but pp j − 1 for all j<i. We can write the order of G as n = p m q r k, where gcd( pq, k) = 1. As we can find an element φ of order p in Aut ((Z q ) i ). Let f : Z p − Aut ((Z q ) i ) be the group homomorphism defined by f (a) = f, where a is a generator of Z p . The semidirect product Z p ⋉(Z q ) i , of Z p and (Z q ) i with respect to f, is a not nilpotent group of order pq i . By assumption on p and q, the group Z p ⋉(Z q ) i has q i Sylow p-subgroup. Therefore, S p (Z p ⋉(Z q ) i ) = q i ( p − 1) and S q (Z p ⋉(Z q ) i ) = q i − 1. Hence Then T is a not nilpotent group of order p m q r . It is easy to see that S p (T) = q i ( p − 1)p m−1 + p m−1 − 1 a n d S q (T) = q r − 1. So In addition Therefore . 0.
Since Z m p × Z r q has the greatest c * (H) among all nilpotent groups H of order d = p m q r , the integer d = p m q r satisfy the property N(d). Lemma 3.4 completes the proof.

Conclusion
This paper determines the minimum value and the maximum value of c * on the set of groups of the same order. More precisely, it is proved that a cyclic group G can be characterized by its order and the value of c * at G. That means, if C is a finite cyclic group, then c * (G) < c * (C) for all noncyclic groups G of the same order as C. On the other hand, it is given in this paper a new characterization, for nilpotent groups, announced as follows: if n be an integer which is not a nilpotent number and max{c * (G) | |G| = n} = c * (K) for some group K of order n, then K is not nilpotent.

Disclosure statement
No potential conflict of interest was reported by the authors.