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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 459135, 4 pages
http://dx.doi.org/10.1155/2014/459135
Research Article

The Schur Multiplier of Pairs of Groups of Order

1Department of Mathematics, Faculty of Science, Islamic Azad University, Firoozkooh Branch, Tehran 3319118651, Iran
2Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Johor Bahru, Johor, Malaysia

Received 3 September 2014; Revised 5 November 2014; Accepted 5 November 2014; Published 25 November 2014

Academic Editor: Kishin Sadarangani

Copyright © 2014 S. Rashid 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.

Abstract

Let be a pair of groups where is a group and is a normal subgroup of . Then the Schur multiplier of pairs of groups is a functorial abelian group . In this paper, for groups of order where and are prime numbers are determined.

1. Introduction

The Schur multiplier was introduced by Schur [1] in 1904. The Schur multiplier of a group , , is isomorphic to in which is a group with a free presentation . He also computed for many different kinds of groups: for example, the dihedral group, metacyclic group, alternating group, and quaternion group. All computations of were then compiled by Karpilovsky [2] in a book entitled “The Schur Multiplier.”

In 1998, Ellis [3] extended the notion of the Schur multiplier of a group to the Schur multiplier of a pair of group, , where is a normal subgroup of . The Schur multiplier of a pair of groups, , is a functorial abelian group whose principal feature is natural exact sequence in which denotes some finiteness-preserving functor from groups to abelian groups (to be precise, is the third homology of a group with integer coefficients). The homomorphisms , , are those due to the functorial of , , and . Ellis [3] also stated that, for any pair of groups, where is the exterior product of and . The exterior product is obtained from by imposing the additional relation for all and the image of a general element in is denoted by for all and .

The nonabelian tensor product, , was introduced by Brown and Loday [4] in 1987. is the group generated by the symbols subject to the relations for all , in and , in . is used in computing the Schur multiplier of the direct product of two groups, . Some computations of the nonabelian tensor product of cyclic group of -power order have been done by Visscher [5] in 1998.

The nonabelian tensor square and Schur multiplier of groups of order , , and has been computed by Jafari et al. [6]. In this paper, the Schur multiplier of pairs of groups of order where and are primes is determined.

In 2007, Moghaddam et al. [7] showed that if is a normal subgroup of such that . In 2012, Rashid et al. [8] determined the commutator subgroups of groups of order . The Schur multiplier, nonabelian tensor square, and capability of groups of order have been considered by Rashid et al. in [9], where and are distinct primes. In [10], they also computed the nonabelian tensor square and capability of groups of order , where is an odd prime.

2. Preliminaries

This section includes some preliminary results that are used in proving our main theorems.

Definition 1 (see [2]). A normal subgroup of is called a normal Hall subgroup of if the order of is coprime to its index in .

Definition 2 (see [2]). is defined as the -stable subgroup of ; that is, where is a subgroup of in which is the semidirect product of a normal subgroup and a subgroup , and is the conjugation of on .

Proposition 3 (see [11]). Let and be distinct primes and let be a finite group of order . Then one of the following holds:(i) and has a normal Sylow -subgroup;(ii) and has a normal Sylow -subgroup;(iii), , , and has a normal 2-subgroup.

Proposition 4 (see [9]). Let be a nonabelian group of order where and are distinct primes. Then exactly one of the following holds:(i) and ;(ii) and ;(iii) and ;(iv) and ;(v) and ;(vi).

Proposition 5 (see [12]). The factor group is abelian. If is a normal subgroup of such that is abelian, then .

Proposition 6 (see [5]). Let and be cyclic groups that act trivially on each other. Then .

Proposition 7 (see [2]). Let be a finite group. Then (i) is a finite group whose elements have order dividing the order of .(ii) if is cyclic.

Proposition 8 (see [2]). If the Sylow -subgroups of are cyclic for all , then .

Proposition 9 (see [2]). Let be a normal Hall subgroup of and a complement of in . Then

Proposition 10 (see [2]). If and are finite groups, then

Proposition 11 (see [6]). Let be a finite nonabelian group. If is a group of order , then

The following propositions are some of the basic results of the Schur multiplier of a pair deduced by Ellis [3], assuming only the existence of the natural exact sequence in (1) and the existence of a certain transfer homomorphism.

Proposition 12 (see [3]). Let ; then .

Proposition 13 (see [3]). Let ; then .

Proposition 14 (see [3]). Suppose that is a finite group. Let the order of the normal subgroup be coprime to its index in and a complement of in . Then and .

3. Main Result

In the following two theorems, the Schur multipliers of pairs of groups of order are stated and proved. We assume that the group is nonabelian.

Theorem 15. Let be a group of order where and are distinct primes, and . If , then the Schur multiplier of pairs of where .

Proof. Let be a group of order where and are distinct primes, and . Since , then by Proposition 3   has a normal Sylow -subgroup: call it . Moreover, so is abelian. Then by Proposition 5, we have ; that is, . Thus, by Proposition 4, or .
Suppose ; then the Schur multiplier of pairs of is computed below.
Case 1. If then by Proposition 11, .
Since , for all normal subgroups of , .
Case 2. If then by Proposition 11, .(i)If then by Proposition 12, .(ii)If then by Proposition 13, . By Proposition 11, .(iii)If then is the semidirect product of and in which and are coprimes, and is a normal Hall subgroup of (refer to Definition 1). Therefore by Proposition 14, since (refer to Proposition 7). Note that, for this case, ; that is, .(iv)If then is nonabelian group of order . (If then by Proposition 5, ; that is, and this statement is a contradiction). Thus the exact sequence shows that where is the kernel of homomorphism to . Then .(v)If , or then by similar way as in (iv), .

Theorem 16. Let be a group of order where and are distinct primes, and . If , then the Schur multiplier of pairs of where .

Proof. Let be a group of order where and are distinct primes, and . Since , has a normal Sylow -subgroup, namely, (refer to Proposition 3). so is abelian. Hence, (refer to Proposition 5); that is, , or . Suppose ; then the Schur multiplier of pairs of is computed below.
(In this case and .)
Case 1. If then and are coprimes. Then, by Definition 1, is a normal Hall subgroup of . Therefore by Proposition 9, where is a complement of and . Thus, (refer to Proposition 7). Hence, (refer to Propositions 10, 6, and 7). (i)If then (refer to Proposition 12).(ii)If then (refer to Proposition 13). Then, .(iii)If then is a normal Hall subgroup of (refer to Definition 1) and is the semidirect product of and in which is a complement of in . Therefore by Proposition 14, since (refer to Proposition 7).(iv)If then is nonabelian group of order . (If then by Proposition 5, ; that is, and this statement is a contradiction). Thus the exact sequence shows that where is the kernel of homomorphism to . Then .(v)If then by similar way as in (iv), .(vi)If or then is abelian group and or but this statement is a contradiction. So when or are not considered.
Case 2. If then . Hence, all Sylow subgroups of are cyclic. Therefore, by Proposition 8, . Thus, for all normal subgroups of , .
Case 3. If then since where is a group of order . By Propositions 7 and 8, and . Thus, for all normal subgroups of , .

4. Conclusion

For a group of order where and are prime numbers, is the unique normal Sylow -subgroups of if , while is the unique normal Sylow -subgroups of if . In this paper, we determined the Schur multiplier of pairs of groups of order . Our proofs show that for groups of order is either 1 or .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Ministry of Education (MOE), Malaysia, and Research Management Centre Universiti Teknologi Malaysia (RMCUTM) for the financial support through the Research University Grant (RUG) Vote no. 04H13. The second author would also like to thank Ministry of Education (MOE), Malaysia, for her MyPhD Scholarship.

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