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.

KEYWORDS:

• Finite groups
• cyclic groups
• nilpotent groups

AMS SUBJECT CLASSIFICATION:

• 20B05
• 20D60

1. Introduction

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

In Section 2, we give the preliminary definitions and results about the functions and . Particularly, we will show that if and are two finite groups, then if and only if the order of and that of are relatively prime, and as a consequence, we will prove that if G is a nilpotent group of order n, then 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 on the set of groups of the same order. More precisely, the main results are:

1. Let G be a finite group. Then for every cyclic group C of the same order as G.

2. Let n be an integer which is not a nilpotent number and for some group K of order n. Then K is not nilpotent.

2. Preliminaries and basic results

This section presents some results and notations that will be useful in the sequel. Given a finite group G, let:

• be the set of all positive divisor of ,

• be the set of all prime divisor of and

• for all .

Definition 2.1:

We define the area of G (the sum of element orders of G) as follows:(2)

Definition 2.2:

We define the prime area of G (the sum of prime element orders of G) as follows:(3)

Example 2.3:

If is a prime number, then

1. ,

2. .

Proposition 2.4:

For every normal N subgroup of G, we have (4)

Proof:

Remark that if x is an element in G such that for some prime number p, then . Hence, or as an element in G/N. That means or . Therefore, Hence, .□

Proposition 2.5:

Let G be a nonabelian group of order where p is a prime number greater than or equal to 3.

2. .

Proof:

It is well known that if G is a nonabelian group of order where p is a prime number greater than or equal to 3, then , and since the dihedral group has 1 element of order 1, p element of order 2, and p−1 element of order p, we obtain and . □

Lemma 2.6:

Let and be two finite groups, then (5)

Proof:

Let then . Therefore, or or Hence, for all prime p, we have(6) where and are, respectively, the identity element of and . Then, we have(7) Hence(8) □

Theorem 2.7:

Let and be two finite groups. Then, the following statements are equivalent:

1. .

2. .

Proof:

Since , for all . Then, is equivalent to . That means or . Hence, for all prime p, we have(9) It follows that(10) Hence(11) Reciprocally, assume that . Using the previous lemma, we obtain(12) Hence, for all prime number p we have or . Consider the contrary that means. Then, there is a prime number p dividing both and . Applying the Cauchy theorem, there are elements and such that . This is a contradiction to or .□

Proposition 2.8:

If 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 and , then(13) Since , we get HK=G. Then, the map defined by is an isomorphism. Applying Theorem 2.7, we obtain(14) □

Theorem 2.9:

Let G be a nilpotent group of order n. Then, the following are equivalent:

1. for every nilpotent group H of order n.

2. Every Sylow subgroup of G has a prime exponent.

Proof:

Put where are distinct primes and 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(15) where is the Sylow -subgroup of H. From Theorem 2.7, we obtain(16) Therefore(17) If every Sylow subgroup of G has a prime exponent, then(18) □

Corollary 2.10:

Let G be a finite group of order where are distinct primes and are positive integers. Then the following statements are equivalent:

1. G is not nilpotent,

3. .

Proof:

The equivalence is a direct consequence of the previous theorem, and the equivalence is proved by Amiri and Jafarian Amiri in [1, Corollary 2.2].□

3. Minimum and maximum values of on the set of groups of the same order

In this section, we investigate the minimum and the maximum value of 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 . Then, is a multiple of .

Proof:

Assume that and . Let for all , the cyclic subgroup of G generated by . Let . Clearly every element in is of order p, and for every i,j, or . So, there exists a subset of such that for all in and(19) Since the set contain elements of order p, we conclude that(20) □

Corollary 3.2:

Let G be a finite group of order , where are distinct primes and are positive integers. Then there exist positive integers such that

Theorem 3.3:

Let G be a finite group. Then 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 elements of order r. Then(21) Using Corollary 3.2, we obtain that

In the following, if d is a positive integer, we say that d satisfy the property if 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 . Then n=ds satisfy the property for all positive integers s such that .

Proof:

Let G be a group of order n=ds such thatIf G is nilpotent, then G can be written as , where and . By hypothesis, there exists a not nilpotent group K of order d such that . Let . Then H is a not nilpotent group of order ds and□

Definition 3.5 ([7]):

A positive integer n is called nilpotent number if every group of order n is nilpotent.

Lemma 3.6 ([7, Theorem 1]:

Let be an integer where are distinct primes and are positive integers. Then n is nilpotent number if and only if for all integers i,j and k with .

Theorem 3.7:

Let n be an integer which is not a nilpotent number. Assume that 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 , and an integer i such that but for all j<i. We can write the order of G as , where . As(22) we can find an element φ of order p in . Let be the group homomorphism defined by , where a is a generator of . The semidirect product , of and with respect to f, is a not nilpotent group of order . By assumption on p and q, the group has Sylow p-subgroup. Therefore, and . Hence(23) Let . Then T is a not nilpotent group of order . It is easy to see that and . So(24) In addition(25) ThereforeSince has the greatest among all nilpotent groups H of order , the integer satisfy the property . Lemma 3.4 completes the proof.

4. Conclusion

This paper determines the minimum value and the maximum value of 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 at G. That means, if C is a finite cyclic group, then 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 for some group K of order n, then K is not nilpotent.

Disclosure statement

No potential conflict of interest was reported by the authors.