The Order Classes of 2-Generator -Groups

Abstract

In order to classify a finite group using its elements orders, the order classes are defined. This partition determines the number of elements for each order. The aim of this paper is to find the order classes of 2-generator -groups of class 2. The results obtained here are supported by Groups, Algorithm and Programming (GAP).

1. Introduction

One of the major partitions for finite groups is the order classes. A basic concept in group theory is that the order of an element denoted by is the smallest positive integer , such that is the identity. The relation “ is of the same order as ” is an equivalence relation, which induces a partition for the group , which is called the order classes. Order classes of symmetric and dihedral groups are completely configured in [1] and [2], respectively.

Clearly, all conjugate elements have the same order. Conjugacy classes are refinement partitions to order classes. Therefore, each order class contains at least one conjugacy class. Du and Shi [3] proved that if a finite group has conjugacy classes number one greater than its same order classes number, then is isomorphic to one of the following groups: , , , , , , , , , or .

In order to classify a finite group using its order classes, there is a new issue obtained by the size of the order classes. That is, a finite group is said to be a perfect order subsets group (POS-group) if the cardinality of each order class divides . Das [4] studied some of the properties of arbitrary POS-groups and constructed a couple of new families of nonabelian POS-groups. He also proved that the alternating group ,  , is not a POS-group. Later, Jones and Toppin [5] proved that any nontrivial finite POS-group has even order.

The classification of all -groups is not completed yet. In 1993 the classification of finite 2-generator -groups of class 2 has been studied in [6]. Ahmad et al. [7] classified 2-generator -groups of class 2 and defined these groups as a central extension of cyclic -groups, that is, to obtain the exact number of conjugacy classes for these groups. In this study we will follow the same classification found in [7], to investigate the order classes of 2-generator -groups of nilpotency class 2.

The results obtained here were found using GAP. Fortunately, using our main theorem, we have developed a practical GAP algorithm to find the order classes of 2-generator -groups of class 2 ( odd prime).

2. Preliminaries and Definitions

Our notation is fairly standard. By we denote the order of a finite group and we denote the identity element of by . The order of an element , denoted by , is the smallest positive integer such that . The set of all possible orders for a finite group will be denoted by . The class of all elements of which have the same order of is called the order class of . Equivalently, the class of all elements of of order is the order class of and is denoted by . The order classes of a group will be denoted by , which consists of all possible pairs of the form for all . The derived subgroup and the center of a group are denoted by and , respectively.

Let be a group. The commutator of is given by . For any subgroups and of a group the commutator subgroup is . Note that the lower central series of a group iswhere for .

Definition 1. A group is called nilpotent if there exists such that , and the smallest such is the class of nilpotency.

All abelian groups are nilpotent of class 1. If is prime, then the group in which every element has order a power of is called a -group. If is a finite -group, then the order of is a power of . Such groups are nilpotent. A group is nilpotent group of class 2 if ; equivalently .

In a finite -group of order , the center is a subgroup of . Using Lagrange’s theorem, it is implied that for some integer .

Lemma 2 (see [8]). Let be a group of nilpotency classes 2. Let and ; then (1),(2),(3),(4).

Lemma 3 (see [6]). Let be a group of nilpotency classes 2 and with and being odd. Then (1),(2).

The following theorem is used to describe the structure of a 2-generator -group of nilpotency class 2 in terms of generators and relations.

Theorem 4 (see [7]). Let be a prime and a positive integer. Every 2-generator -group of order and class 2 corresponds to an ordered 5-tuple of integers, , such that (1),(2),(3) and , where corresponds to the group presented byMoreover (1)if , then is isomorphic to(a) when ;(b) when or ;(c) when ;(2)if , or and , then is isomorphic to ;(3)if and , then is isomorphic to(a) when ;(b) when ;(c) when and .The groups listed in 1(a)–3(c) are pairwise nonisomorphic.

If is prime and is a 2-generator -group of class 2, with , , then , where [7]. Let be a 2-generator -group of class 2. Then , , and are the polycyclic series of . Hence, and are the polycyclic generators of . Therefore, if , then can be written uniquely as , where , , and .

3. Order Classes of 2-Generator -Groups of Nilpotency Class 2

The previous classification for 2-generator -groups will be used to obtain the order classes of these groups. Let be the set of all 2-generator -groups of nilpotency class 2 with being an odd prime and , , . To find the order classes of a group , we need to answer some important issues related to , such as the description of the available orders ; the largest possible order , to achieve the set ; the count of elements of each order family to obtain . The following lemmas will justify these issues and concepts to establish the order classes in terms of .

Lemma 5. Let be the group generated by and , with , . Then , where .

Proof. The proof follows directly using Lemma 3, since

Reasonably, for , the order 5-tuple of integers in Theorem 4 was configured to construct the group . But the new order pair obtained by the generators orders is a different pair; it is clear that for all . So that will never be used instead of , although they are occasionally similar.

Let . Then the order of any element in should divides . Therefore, if , then should be written as a power of . Thus, where (if , then is cyclic group). The following lemma establishes the largest possible order in terms of the generators orders .

Lemma 6. If is the group generated by and , such that , , letting , then the exponent of , denoted by , is given by

Proof. Let and . Theorem 4 gives that , where , , and . ThereforeThenNotice that . Hence

Using Lemma 6, it follows that the set of all possible orders is , where . Hence , where is the number of elements of order for .

According to the previous classifications our main results will be as follows.

Theorem 7. Let be the group generated by and , with , , , and    for all . Let . Then, has elements of order , , wheresuch that (1)  (2)  (3)If . Then .(4)If . Then for .(5).

Proof. The identity element is the only element in of order 1; therefore . Without loss of generality, let .
() Let ; then , where , , and .
Using (6), it is implied thatSince , hence ; thereforeThenCase 1. If , thenHence, there are choices for ; they are originally for and similarly there are choices for . Therefore there are choices for . Note that for and . ThenCase 2. If , thenTherefore, there are choices for . The identity element is omitted. Thus() Using similar arguments as Case , thenIf , thenHence, there are choices for and choices for . Hence, ; else, . ThenThere are choices for and choices for , implying that .
() Similarly, if , thenThenHence, .
() If , for all , then the number of choices for reduces in a ratio of for each . Thus .
() When , thenHence, .

Corollary 8. Let be the group generated by and , with , . Then is not a POS-group.

Proof. It is enough to show that there exists such that . For , wheresuppose, on the contrary, that . Then there exists with and . ThereforeThenso that . If , then , which implies that , a contradiction. If , then and have no solution for as an integer which gives a contradiction as well. It follows that there is no integer such that . Thus , which means that is not a POS-group.

4. GAP

This study includes GAP’s algorithms. Algorithm 1 (see the appendix) is derived from Theorem 7 and is used to find the order classes of all 2-generator -groups of nilpotency class 2 (as a list), by determining the values of and . Algorithm 2 (see the appendix) is being built using the ordinary GAP formulas and commands (already installed with GAP’s packages) to give the same results as Algorithm 1.

Example 9. When both Algorithms 1 and 2 are used to find the order classes for all 2-generator -groups of class 2, where and , the results obtained are as follows:

Similarly, for and

The time required for Algorithm 2 to find the order classes of 2-generator -groups of class 2, when and , is 38064 milliseconds while Algorithm 1 needs 14180 milliseconds to find the same results. Next, for and , Algorithm 2 could not complete the process, for the group size (4879681) exceeded the permitted memory size. Conversely, Algorithm 1 takes 100355 milliseconds. Distinctly, Algorithm 1 is much better than the ordinary GAP algorithm and it can be used instead.

5. Conclusion

In this paper, the classification of 2-generator -groups of nilpotency class 2 has been used to determine the order classes of this type of groups. This work contains an appreciable number of imperative results. We have used these results to create a GAP algorithm (Algorithm 1) to find the order classes of 2-generator -groups of nilpotency class 2, odd prime. When Algorithm 1 is compared to Algorithm 2, which has been used for the same purpose, we have found that Algorithm 1 does not use all of the group elements and only depends on two elements (generators) to classify the order class of this group, while Algorithm 2 uses all of the group elements to give the same results. Therefore, it works very slow and interrupts large size groups, on the contrary to Algorithm 1.

Appendix

See Algorithms 1 and 2.

Competing Interests

The authors declare that they have no competing interests.

References

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Copyright

Copyright © 2016 Mahmoud Bashir Alhasanat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.