Research Article - science.utm.my of Mathematics, Faculty of Science, Islamic Azad University,...
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Research Article The Schur Multiplier of Pairs of Groups of Order 2 S. Rashid, 1 A. A. Nawi, 2 N. M. Mohd Ali, 2 and N. H. Sarmin 2 1 Department of Mathematics, Faculty of Science, Islamic Azad University, Firoozkooh Branch, Tehran 3319118651, Iran 2 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Johor Bahru, Johor, Malaysia Correspondence should be addressed to S. Rashid; [email protected] 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. is 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. Let (, ) be a pair of groups where is a group and is a normal subgroup of . en the Schur multiplier of pairs of groups (, ) is a functorial abelian group (, ). In this paper, (, ) for groups of order 2 where and are prime numbers are determined. 1. Introduction e Schur multiplier was introduced by Schur [1] in 1904. e Schur multiplier of a group , (), is isomorphic to ∩[, ]/[, ] in which is a group with a free presentation 1 → → → → 1. 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 “e 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 . e Schur multiplier of a pair of groups, (, ), is a functorial abelian group (, ) whose principal feature is natural exact sequence 3 () → 3 ( ) → (,) → () → ( ) → [, ] → () → ( ) → 1, (1) in which 3 (−) denotes some finiteness-preserving functor from groups to abelian groups (to be precise, 3 (−) is the third homology of a group with integer coefficients). e homomorphisms , , are those due to the functorial of 3 (−), (−), and (−) . Ellis [3] also stated that, for any pair (, ) of groups, (, ) ≅ ker (∧ → ) where ∧ is the exterior product of and . e exterior product ∧ is obtained from ⊗ by imposing the additional relation ⊗=1 for all (, ) ∈ ∧ and the image of a general element ⊗ in ∧ is denoted by ∧ for all ∈ and ∈. e nonabelian tensor product, ⊗, was introduced by Brown and Loday [4] in 1987. ⊗ is the group generated by the symbols ⊗ℎ subject to the relations ⊗ℎ=( ⊗ ℎ) ( ⊗ ℎ), ⊗ ℎℎ = ( ⊗ ℎ) ( ℎ ⊗ ℎ ℎ ), (2) for all , in and ℎ, ℎ 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. e nonabelian tensor square and Schur multiplier of groups of order 2 , 2 , and 2 has been computed by Jafari et al. [6]. In this paper, the Schur multiplier of pairs of groups of order 2 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 8. e Schur multiplier, nonabelian tensor square, and capability of groups of order 2 have been considered by Rashid et al. in [9], where Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 459135, 4 pages http://dx.doi.org/10.1155/2014/459135
Transcript of Research Article - science.utm.my of Mathematics, Faculty of Science, Islamic Azad University,...
Research ArticleThe Schur Multiplier of Pairs of Groups of Order 1199012119902
S Rashid1 A A Nawi2 N M Mohd Ali2 and N H Sarmin2
1 Department of Mathematics Faculty of Science Islamic Azad University Firoozkooh Branch Tehran 3319118651 Iran2Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia (UTM) 81310 Johor Bahru Johor Malaysia
Correspondence should be addressed to S Rashid samadrashid47yahoocom
Received 3 September 2014 Revised 5 November 2014 Accepted 5 November 2014 Published 25 November 2014
Academic Editor Kishin Sadarangani
Copyright copy 2014 S Rashid et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Let (119866119873) be a pair of groups where 119866 is a group and 119873 is a normal subgroup of 119866 Then the Schur multiplier of pairs of groups(119866119873) is a functorial abelian group119872(119866119873) In this paper119872(119866119873) for groups of order 1199012119902 where 119901 and 119902 are prime numbers aredetermined
1 Introduction
The Schur multiplier was introduced by Schur [1] in 1904The Schur multiplier of a group 119866 119872(119866) is isomorphic to119877cap[119865 119865][119877 119865] in which119866 is a groupwith a free presentation1 rarr 119877 rarr 119865 rarr 119866 rarr 1 He also computed 119872(119866)
for many different kinds of groups for example the dihedralgroup metacyclic group alternating group and quaterniongroup All computations of 119872(119866) were then compiled byKarpilovsky [2] in a book entitled ldquoThe Schur Multiplierrdquo
In 1998 Ellis [3] extended the notion of the Schurmultiplier of a group to the Schur multiplier of a pair ofgroup (119866119873) where 119873 is a normal subgroup of 119866 TheSchur multiplier of a pair of groups (119866119873) is a functorialabelian group 119872(119866119873) whose principal feature is naturalexact sequence
1198673(119866)120578
997888rarr 1198673(119866
119873) 997888rarr 119872(119866119873) 997888rarr 119872(119866)
120583
997888rarr 119872(119866
119873)
997888rarr119873
[119873119866]997888rarr (119866)
119886119887 120572
997888rarr (119866
119873)
119886119887
997888rarr 1
(1)
in which 1198673(minus) denotes some finiteness-preserving functor
from groups to abelian groups (to be precise 1198673(minus) is the
third homology of a group with integer coefficients) Thehomomorphisms 120578 120583 120572 are those due to the functorial of1198673(minus)119872(minus) and (minus)
119886119887 Ellis [3] also stated that for any pair(119866119873) of groups119872(119866119873) cong ker (119873and119866 rarr 119866)where119873and119866
is the exterior product of119873 and119866The exterior product119873and119866
is obtained from 119873 otimes 119866 by imposing the additional relation119899 otimes 119892 = 1 for all (119899 119892) isin 119873 and 119866 and the image of a generalelement 119899 otimes 119892 in119873 and 119866 is denoted by 119899 and 119892 for all 119899 isin 119873 and119892 isin 119866
The nonabelian tensor product119866otimes119867 was introduced byBrown and Loday [4] in 1987119866otimes119867 is the group generated bythe symbols 119892 otimes ℎ subject to the relations
1198921198921015840otimes ℎ = (
1198921198921015840otimes119892
ℎ) (119892 otimes ℎ)
119892 otimes ℎℎ1015840= (119892 otimes ℎ) (
ℎ119892 otimesℎ
ℎ
1015840
)
(2)
for all1198921198921015840 in119866 and ℎ ℎ1015840 in119867119866otimes119867 is used in computing theSchur multiplier of the direct product of two groups119872(119866 times
119867) Some computations of the nonabelian tensor product ofcyclic group of 119901-power order have been done by Visscher [5]in 1998
The nonabelian tensor square and Schur multiplier ofgroups of order 1199012119902 1199011199022 and 119901
2119902119903 has been computed by
Jafari et al [6] In this paper the Schur multiplier of pairs ofgroups of order 1199012119902 where 119901 and 119902 are primes is determined
In 2007 Moghaddam et al [7] showed that 119872(119866119873) cong
119877cap [119878 119865][119877 119865] if 119878 is a normal subgroup of 119865 such that119873 cong
119878119877 In 2012 Rashid et al [8] determined the commutatorsubgroups of groups of order 8119902 The Schur multipliernonabelian tensor square and capability of groups of order1199012119902 have been considered by Rashid et al in [9] where 119901
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 459135 4 pageshttpdxdoiorg1011552014459135
2 Mathematical Problems in Engineering
and 119902 are distinct primes In [10] they also computed thenonabelian tensor square and capability of groups of order8119902 where 119902 is an odd prime
2 Preliminaries
This section includes some preliminary results that are usedin proving our main theorems
Definition 1 (see [2]) A normal subgroup 119873 of 119866 is called anormal Hall subgroup of 119866 if the order of119873 is coprime to itsindex in 119866
Definition 2 (see [2]) 119872(119873)119879 is defined as the 119879-stable
subgroup of119872(119873) that is119872(119873)119879= 119891 isin 119872(119873)|Con119905
119873(119891) =
119891 for all 119905 isin 119879 where 119879 is a subgroup of 119866 in which 119866 is thesemidirect product of a normal subgroup 119873 and a subgroup119879 and Con119905
119873(119891) is the conjugation of 119905 on 119891
Proposition 3 (see [11]) Let 119901 and 119902 be distinct primes andlet 119866 be a finite group of order 1199012119902 Then one of the followingholds
(i) 119901 gt 119902 and 119866 has a normal Sylow 119901-subgroup(ii) 119901 lt 119902 and 119866 has a normal Sylow 119902-subgroup(iii) 119901 = 2 119902 = 3119866 cong 119860
4 and119866 has a normal 2-subgroup
Proposition 4 (see [9]) Let 119866 be a nonabelian group of order1199012119902 where 119901 and 119902 are distinct primes Then exactly one of the
following holds
(i) 1198661015840 cong Z119901and 119866
119886119887cong Z119901119902
(ii) 1198661015840 cong Z1199012 and 119866
119886119887cong Z119902
(iii) 1198661015840 cong Z119901times Z119901and 119866
119886119887cong Z119902
(iv) 1198661015840 cong Z119902and 119866
119886119887cong Z1199012
(v) 1198661015840 cong Z119902and 119866
119886119887cong Z119901times Z119901
(vi) 1198661015840 cong Z2times Z2
Proposition 5 (see [12]) The factor group 1198661198661015840 is abelian If
119870 is a normal subgroup of 119866 such that 119866119870 is abelian then1198661015840sube 119870
Proposition 6 (see [5]) Let 119866 cong Z119898and 119867 cong Z
119899be cyclic
groups that act trivially on each other Then 119866 otimes 119867 cong Z(119898119899)
Proposition 7 (see [2]) Let 119866 be a finite group Then(i) 119872(119866) is a finite group whose elements have order
dividing the order of 119866(ii) 119872(119866) = 1 if 119866 is cyclic
Proposition 8 (see [2]) If the Sylow 119901-subgroups of 119866 arecyclic for all 119901 | |119866| then119872(119866) = 1
Proposition 9 (see [2]) Let 119873 be a normal Hall subgroup of119866 and 119879 a complement of119873 in 119866 Then
119872(119866) cong 119872(119879) times119872 (119873)119879 (3)
Proposition 10 (see [2]) If 1198661and 119866
2are finite groups then
119872(1198661times 1198662) = 119872(119866
1) times 119872(119866
2) times (119866
1otimes 1198662) (4)
Proposition 11 (see [6]) Let119866 be a finite nonabelian group If119866 is a group of order 1199012119902 then
119872(119866) =
1 119894119891 1198661015840= Z119902 119866119886119887
= Z1199012
Z119901 119894119891 119866
1015840= Z119902 119866119886119887
= Z119901times Z119901
Z2 119894119891 119866
119886119887= Z2times Z2
(5)
The following propositions are some of the basic results ofthe Schur multiplier of a pair deduced by Ellis [3] assumingonly the existence of the natural exact sequence in (1) and theexistence of a certain transfer homomorphism
Proposition 12 (see [3]) Let119873 = 1 then119872(119866119873) = 1
Proposition 13 (see [3]) Let119873 = 119866 then119872(119866119866) = 119872(119866)
Proposition 14 (see [3]) Suppose that 119866 is a finite group Letthe order of the normal subgroup 119873 be coprime to its indexin 119866 and 119879 a complement of 119873 in 119866 Then 119866 cong 119873 ⋊ 119879 and119872(119866119873) cong 119872(119873)
119879
3 Main Result
In the following two theorems the Schur multipliers of pairsof groups of order 1199012119902 are stated and proved We assume thatthe group is nonabelian
Theorem 15 Let 119866 be a group of order 1199012119902 where 119901 and 119902 aredistinct primes and 119901 lt 119902 If119873 ⊲ 119866 then the Schur multiplierof pairs of 119866
119872(119866119873) =
1 119894119891 119866119886119887
cong Z1199012 119900119903 119866
119886119887cong Z119901times Z119901
119908ℎ119890119899 119873 = 1 119900119903 Z119902
Z119901 119894119891 119866
119886119887cong Z119901times Z119901
119908ℎ119890119899 119873 = 119866Z119901Z119901119902Z119901times Z119901119900119903 Z1199012
(6)
where 119866119886119887 = 1198661198661015840
Proof Let119866be a groupof order1199012119902where119901 and 119902 are distinctprimes and 119901 lt 119902 Since 119901 lt 119902 then by Proposition 3 119866 hasa normal Sylow 119902-subgroup call it119876 Moreover [119866 119876] = 119901
2
so 119866119876 is abelian Then by Proposition 5 we have 1198661015840sube 119876
that is 1198661015840 = Z119902 Thus by Proposition 4 119866119886119887 cong Z
1199012 or 119866119886119887 cong
Z119901times Z119901
Suppose119873 ⊲ 119866 then the Schur multiplier of pairs of 119866 iscomputed below
Case 1 If 119866119886119887 cong Z1199012 then by Proposition 11119872(119866) = 1
Since 119872(119866) = 1 for all normal subgroups 119873 of 119866119872(119866119873) le 119872(119866) = 1
Mathematical Problems in Engineering 3
Case 2 If 119866119886119887 cong Z119901timesZ119901then by Proposition 11119872(119866) = Z
119901
(i) If 119873 = 1 then by Proposition 12 119872(119866119873) =
119872(119866 1) = 1(ii) If 119873 = 119866 then by Proposition 13 119872(119866119873) =
119872(119866 119866) = 119872(119866) By Proposition 11119872(119866) = Z119901
(iii) If119873 = Z119902then 119866 is the semidirect product ofZ
119902and
119867 in which |119873| and [119866 119873] are coprimes and 119873 isa normal Hall subgroup of 119866 (refer to Definition 1)Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since 119872(Z119902) = 1 (refer to Proposition 7) Note that
for this case 119866119873 = 1198661198661015840
= Z1199012 that is 119866119873 = Z
1199012
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 thatis Z119902sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If119873 = Z119901119902 Z1199012 or Z
119901timesZ119901then by similar way as in
(iv)119872(119866119873) = Z119901
Theorem 16 Let 119866 be a group of order 1199012119902 where 119901 and 119902 aredistinct primes and 119901 gt 119902 If119873 ⊲ 119866 then the Schur multiplierof pairs of 119866
119872(119866119873)
=
1 119894119891 1198661015840cong Z119901119900119903 1198661015840cong Z1199012 119900119903 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 1 119900119903 Z119902
Z119901 119894119891 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 119866 Z119901119900119903 Z119901times Z119901
(7)
where 119866119886119887 = 1198661198661015840
Proof Let119866be a groupof order1199012119902where119901 and 119902 are distinctprimes and 119901 gt 119902 Since 119901 gt 119902 119866 has a normal Sylow 119901-subgroup namely 119875 (refer to Proposition 3) [119866 119875] = 119902 so119866119875 is abelian Hence 1198661015840 sube 119875 (refer to Proposition 5) thatis 1198661015840 cong Z
119901times Z119901 Z1199012 or Z
119901 Suppose119873 ⊲ 119866 then the Schur
multiplier of pairs of 119866 is computed below(In this case119873 = 1Z
119902Z119901Z119901times Z119901and 119866)
Case 1 If 1198661015840 cong Z119901times Z119901then |119866
1015840| = 1199012 and [119866 119866
1015840] = 119902 are
coprimesThen byDefinition 11198661015840 is a normalHall subgroupof 119866 Therefore by Proposition 9 119872(119866) = 119872(119879) times 119872(119866
1015840)119879
where 119879 is a complement of 1198661015840 and 119879 cong Z119902 Thus 119872(119866) =
119872(119879) times 119872(Z119901times Z119901)119879 119872(119879) = 1 (refer to Proposition 7)
Hence119872(119866) = Z119901(refer to Propositions 10 6 and 7)
(i) If 119873 = 1 then 119872(119866119873) = 119872(119866 1) = 1 (refer toProposition 12)
(ii) If119873 = 119866 then119872(119866119873) = 119872(119866 119866) = 119872(119866) (refer toProposition 13) Then119872(119866) = Z
119901
(iii) If 119873 = Z119902then 119873 is a normal Hall subgroup of 119866
(refer to Definition 1) and119866 is the semidirect productof 119873 and 119867 in which 119867 is a complement of 119873 in 119866Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since119872(Z119902) = 1 (refer to Proposition 7)
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 that isZ119901times Z119901sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If 119873 = Z119901times Z119901then by similar way as in (iv)
119872(119866119873) = Z119901
(vi) If 119873 = Z119901119902
or Z1199012 then 119866119873 is abelian group and
1198661015840cong Z119901times Z119901sube 119873 cong Z
119901119902or Z1199012 but this statement
is a contradiction So119872(119866119873) when119873 = Z119901119902
orZ1199012
are not considered
Case 2 If 1198661015840 cong Z1199012 then 119866119866
1015840cong 119866119886119887
cong Z119902 Hence all
Sylow subgroups of 119866 are cyclic Therefore by Proposition 8119872(119866) = 1Thus for all normal subgroups119873 of119866119872(119866119873) le
119872(119866) = 1
Case 3 If 1198661015840 cong Z119901then 119872(119866) = 1 since 119872(119866) = 119872(119866
1015840) times
119872(119870) where119870 is a group of order 119901119902 By Propositions 7 and8119872(119866
1015840) = 1 and119872(119870) = 1 Thus for all normal subgroups
119873 of 119866119872(119866119873) le 119872(119866) = 1
4 Conclusion
For a group119866 of order 1199012119902where 119901 and 119902 are prime numbers119876 is the unique normal Sylow 119902-subgroups of 119866 if 119901 lt 119902while 119875 is the unique normal Sylow 119901-subgroups of 119866 if 119901 gt
119902 In this paper we determined the Schur multiplier of pairsof groups of order 1199012119902 Our proofs show that 119872(119866119873) forgroups of order 1199012119902 is either 1 or Z
119901
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Ministry of Education(MOE)Malaysia and ResearchManagement Centre Univer-siti TeknologiMalaysia (RMCUTM) for the financial supportthrough the Research University Grant (RUG) Vote no04H13 The second author would also like to thank Ministryof Education (MOE) Malaysia for her MyPhD Scholarship
References
[1] I Schur ldquoUber die Darstellung der endlichen gruppen durchgebrochen lineare substitutionenrdquo Journal fur die Reine und
4 Mathematical Problems in Engineering
Angewandte Mathematik vol 1904 no 127 pp 20ndash50 19041904
[2] G KarpilovskyThe Schur Multiplier Clarendon Press OxfordUK 1987
[3] G Ellis ldquoThe Schur multiplier of a pair of groupsrdquo AppliedCategorical Structures vol 6 no 3 pp 355ndash371 1998
[4] R Brown and J-L Loday ldquoVan Kampen theorems for diagramsof spacesrdquo Topology vol 26 no 3 pp 311ndash335 1987
[5] M Visscher On the nonabelian tensor products of groups [PhDdissertation] State University of NewYork at Binghamton 1998
[6] S H Jafari P Niroomand and A Erfanian ldquoThe nonabeliantensor square and Schur multiplier of groups of order 1199012119902 1199011199022and 119901
2119902119903rdquo Algebra Colloquium vol 10 pp 1083ndash1088 2012
[7] M R R Moghaddam A R Salemkar and K Chiti ldquoSomeproperties on the Schur multiplier of a pair of groupsrdquo Journalof Algebra vol 312 no 1 pp 1ndash8 2007
[8] S Rashid N H Sarmin A Erfanian and N M Mohd Ali ldquoOnthe commutator subgroups of groups of order 8qrdquo Journal ofComputer Science amp Computational Mathematics vol 2 no 2pp 5ndash7 2012
[9] S Rashid N H Sarmin A Erfanian and N M Ali ldquoOn thenonabelian tensor square and capability of groups of order p2qrdquoArchiv der Mathematik vol 97 no 4 pp 299ndash306 2011
[10] S Rashid N H Sarmin A Erfanian N M Ali and R ZainalldquoOn the nonabelian tensor square and capability of groups oforder 8119902rdquo Indagationes Mathematicae New Series vol 24 no 3pp 581ndash588 2013
[11] M Quick MT5824 Topics in Groups Lecture Notes Universityof St Andrews 2004ndash2009
[12] M HallTheTheory of Groups Macmillan New York NY USA1959
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
and 119902 are distinct primes In [10] they also computed thenonabelian tensor square and capability of groups of order8119902 where 119902 is an odd prime
2 Preliminaries
This section includes some preliminary results that are usedin proving our main theorems
Definition 1 (see [2]) A normal subgroup 119873 of 119866 is called anormal Hall subgroup of 119866 if the order of119873 is coprime to itsindex in 119866
Definition 2 (see [2]) 119872(119873)119879 is defined as the 119879-stable
subgroup of119872(119873) that is119872(119873)119879= 119891 isin 119872(119873)|Con119905
119873(119891) =
119891 for all 119905 isin 119879 where 119879 is a subgroup of 119866 in which 119866 is thesemidirect product of a normal subgroup 119873 and a subgroup119879 and Con119905
119873(119891) is the conjugation of 119905 on 119891
Proposition 3 (see [11]) Let 119901 and 119902 be distinct primes andlet 119866 be a finite group of order 1199012119902 Then one of the followingholds
(i) 119901 gt 119902 and 119866 has a normal Sylow 119901-subgroup(ii) 119901 lt 119902 and 119866 has a normal Sylow 119902-subgroup(iii) 119901 = 2 119902 = 3119866 cong 119860
4 and119866 has a normal 2-subgroup
Proposition 4 (see [9]) Let 119866 be a nonabelian group of order1199012119902 where 119901 and 119902 are distinct primes Then exactly one of the
following holds
(i) 1198661015840 cong Z119901and 119866
119886119887cong Z119901119902
(ii) 1198661015840 cong Z1199012 and 119866
119886119887cong Z119902
(iii) 1198661015840 cong Z119901times Z119901and 119866
119886119887cong Z119902
(iv) 1198661015840 cong Z119902and 119866
119886119887cong Z1199012
(v) 1198661015840 cong Z119902and 119866
119886119887cong Z119901times Z119901
(vi) 1198661015840 cong Z2times Z2
Proposition 5 (see [12]) The factor group 1198661198661015840 is abelian If
119870 is a normal subgroup of 119866 such that 119866119870 is abelian then1198661015840sube 119870
Proposition 6 (see [5]) Let 119866 cong Z119898and 119867 cong Z
119899be cyclic
groups that act trivially on each other Then 119866 otimes 119867 cong Z(119898119899)
Proposition 7 (see [2]) Let 119866 be a finite group Then(i) 119872(119866) is a finite group whose elements have order
dividing the order of 119866(ii) 119872(119866) = 1 if 119866 is cyclic
Proposition 8 (see [2]) If the Sylow 119901-subgroups of 119866 arecyclic for all 119901 | |119866| then119872(119866) = 1
Proposition 9 (see [2]) Let 119873 be a normal Hall subgroup of119866 and 119879 a complement of119873 in 119866 Then
119872(119866) cong 119872(119879) times119872 (119873)119879 (3)
Proposition 10 (see [2]) If 1198661and 119866
2are finite groups then
119872(1198661times 1198662) = 119872(119866
1) times 119872(119866
2) times (119866
1otimes 1198662) (4)
Proposition 11 (see [6]) Let119866 be a finite nonabelian group If119866 is a group of order 1199012119902 then
119872(119866) =
1 119894119891 1198661015840= Z119902 119866119886119887
= Z1199012
Z119901 119894119891 119866
1015840= Z119902 119866119886119887
= Z119901times Z119901
Z2 119894119891 119866
119886119887= Z2times Z2
(5)
The following propositions are some of the basic results ofthe Schur multiplier of a pair deduced by Ellis [3] assumingonly the existence of the natural exact sequence in (1) and theexistence of a certain transfer homomorphism
Proposition 12 (see [3]) Let119873 = 1 then119872(119866119873) = 1
Proposition 13 (see [3]) Let119873 = 119866 then119872(119866119866) = 119872(119866)
Proposition 14 (see [3]) Suppose that 119866 is a finite group Letthe order of the normal subgroup 119873 be coprime to its indexin 119866 and 119879 a complement of 119873 in 119866 Then 119866 cong 119873 ⋊ 119879 and119872(119866119873) cong 119872(119873)
119879
3 Main Result
In the following two theorems the Schur multipliers of pairsof groups of order 1199012119902 are stated and proved We assume thatthe group is nonabelian
Theorem 15 Let 119866 be a group of order 1199012119902 where 119901 and 119902 aredistinct primes and 119901 lt 119902 If119873 ⊲ 119866 then the Schur multiplierof pairs of 119866
119872(119866119873) =
1 119894119891 119866119886119887
cong Z1199012 119900119903 119866
119886119887cong Z119901times Z119901
119908ℎ119890119899 119873 = 1 119900119903 Z119902
Z119901 119894119891 119866
119886119887cong Z119901times Z119901
119908ℎ119890119899 119873 = 119866Z119901Z119901119902Z119901times Z119901119900119903 Z1199012
(6)
where 119866119886119887 = 1198661198661015840
Proof Let119866be a groupof order1199012119902where119901 and 119902 are distinctprimes and 119901 lt 119902 Since 119901 lt 119902 then by Proposition 3 119866 hasa normal Sylow 119902-subgroup call it119876 Moreover [119866 119876] = 119901
2
so 119866119876 is abelian Then by Proposition 5 we have 1198661015840sube 119876
that is 1198661015840 = Z119902 Thus by Proposition 4 119866119886119887 cong Z
1199012 or 119866119886119887 cong
Z119901times Z119901
Suppose119873 ⊲ 119866 then the Schur multiplier of pairs of 119866 iscomputed below
Case 1 If 119866119886119887 cong Z1199012 then by Proposition 11119872(119866) = 1
Since 119872(119866) = 1 for all normal subgroups 119873 of 119866119872(119866119873) le 119872(119866) = 1
Mathematical Problems in Engineering 3
Case 2 If 119866119886119887 cong Z119901timesZ119901then by Proposition 11119872(119866) = Z
119901
(i) If 119873 = 1 then by Proposition 12 119872(119866119873) =
119872(119866 1) = 1(ii) If 119873 = 119866 then by Proposition 13 119872(119866119873) =
119872(119866 119866) = 119872(119866) By Proposition 11119872(119866) = Z119901
(iii) If119873 = Z119902then 119866 is the semidirect product ofZ
119902and
119867 in which |119873| and [119866 119873] are coprimes and 119873 isa normal Hall subgroup of 119866 (refer to Definition 1)Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since 119872(Z119902) = 1 (refer to Proposition 7) Note that
for this case 119866119873 = 1198661198661015840
= Z1199012 that is 119866119873 = Z
1199012
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 thatis Z119902sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If119873 = Z119901119902 Z1199012 or Z
119901timesZ119901then by similar way as in
(iv)119872(119866119873) = Z119901
Theorem 16 Let 119866 be a group of order 1199012119902 where 119901 and 119902 aredistinct primes and 119901 gt 119902 If119873 ⊲ 119866 then the Schur multiplierof pairs of 119866
119872(119866119873)
=
1 119894119891 1198661015840cong Z119901119900119903 1198661015840cong Z1199012 119900119903 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 1 119900119903 Z119902
Z119901 119894119891 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 119866 Z119901119900119903 Z119901times Z119901
(7)
where 119866119886119887 = 1198661198661015840
Proof Let119866be a groupof order1199012119902where119901 and 119902 are distinctprimes and 119901 gt 119902 Since 119901 gt 119902 119866 has a normal Sylow 119901-subgroup namely 119875 (refer to Proposition 3) [119866 119875] = 119902 so119866119875 is abelian Hence 1198661015840 sube 119875 (refer to Proposition 5) thatis 1198661015840 cong Z
119901times Z119901 Z1199012 or Z
119901 Suppose119873 ⊲ 119866 then the Schur
multiplier of pairs of 119866 is computed below(In this case119873 = 1Z
119902Z119901Z119901times Z119901and 119866)
Case 1 If 1198661015840 cong Z119901times Z119901then |119866
1015840| = 1199012 and [119866 119866
1015840] = 119902 are
coprimesThen byDefinition 11198661015840 is a normalHall subgroupof 119866 Therefore by Proposition 9 119872(119866) = 119872(119879) times 119872(119866
1015840)119879
where 119879 is a complement of 1198661015840 and 119879 cong Z119902 Thus 119872(119866) =
119872(119879) times 119872(Z119901times Z119901)119879 119872(119879) = 1 (refer to Proposition 7)
Hence119872(119866) = Z119901(refer to Propositions 10 6 and 7)
(i) If 119873 = 1 then 119872(119866119873) = 119872(119866 1) = 1 (refer toProposition 12)
(ii) If119873 = 119866 then119872(119866119873) = 119872(119866 119866) = 119872(119866) (refer toProposition 13) Then119872(119866) = Z
119901
(iii) If 119873 = Z119902then 119873 is a normal Hall subgroup of 119866
(refer to Definition 1) and119866 is the semidirect productof 119873 and 119867 in which 119867 is a complement of 119873 in 119866Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since119872(Z119902) = 1 (refer to Proposition 7)
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 that isZ119901times Z119901sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If 119873 = Z119901times Z119901then by similar way as in (iv)
119872(119866119873) = Z119901
(vi) If 119873 = Z119901119902
or Z1199012 then 119866119873 is abelian group and
1198661015840cong Z119901times Z119901sube 119873 cong Z
119901119902or Z1199012 but this statement
is a contradiction So119872(119866119873) when119873 = Z119901119902
orZ1199012
are not considered
Case 2 If 1198661015840 cong Z1199012 then 119866119866
1015840cong 119866119886119887
cong Z119902 Hence all
Sylow subgroups of 119866 are cyclic Therefore by Proposition 8119872(119866) = 1Thus for all normal subgroups119873 of119866119872(119866119873) le
119872(119866) = 1
Case 3 If 1198661015840 cong Z119901then 119872(119866) = 1 since 119872(119866) = 119872(119866
1015840) times
119872(119870) where119870 is a group of order 119901119902 By Propositions 7 and8119872(119866
1015840) = 1 and119872(119870) = 1 Thus for all normal subgroups
119873 of 119866119872(119866119873) le 119872(119866) = 1
4 Conclusion
For a group119866 of order 1199012119902where 119901 and 119902 are prime numbers119876 is the unique normal Sylow 119902-subgroups of 119866 if 119901 lt 119902while 119875 is the unique normal Sylow 119901-subgroups of 119866 if 119901 gt
119902 In this paper we determined the Schur multiplier of pairsof groups of order 1199012119902 Our proofs show that 119872(119866119873) forgroups of order 1199012119902 is either 1 or Z
119901
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Ministry of Education(MOE)Malaysia and ResearchManagement Centre Univer-siti TeknologiMalaysia (RMCUTM) for the financial supportthrough the Research University Grant (RUG) Vote no04H13 The second author would also like to thank Ministryof Education (MOE) Malaysia for her MyPhD Scholarship
References
[1] I Schur ldquoUber die Darstellung der endlichen gruppen durchgebrochen lineare substitutionenrdquo Journal fur die Reine und
4 Mathematical Problems in Engineering
Angewandte Mathematik vol 1904 no 127 pp 20ndash50 19041904
[2] G KarpilovskyThe Schur Multiplier Clarendon Press OxfordUK 1987
[3] G Ellis ldquoThe Schur multiplier of a pair of groupsrdquo AppliedCategorical Structures vol 6 no 3 pp 355ndash371 1998
[4] R Brown and J-L Loday ldquoVan Kampen theorems for diagramsof spacesrdquo Topology vol 26 no 3 pp 311ndash335 1987
[5] M Visscher On the nonabelian tensor products of groups [PhDdissertation] State University of NewYork at Binghamton 1998
[6] S H Jafari P Niroomand and A Erfanian ldquoThe nonabeliantensor square and Schur multiplier of groups of order 1199012119902 1199011199022and 119901
2119902119903rdquo Algebra Colloquium vol 10 pp 1083ndash1088 2012
[7] M R R Moghaddam A R Salemkar and K Chiti ldquoSomeproperties on the Schur multiplier of a pair of groupsrdquo Journalof Algebra vol 312 no 1 pp 1ndash8 2007
[8] S Rashid N H Sarmin A Erfanian and N M Mohd Ali ldquoOnthe commutator subgroups of groups of order 8qrdquo Journal ofComputer Science amp Computational Mathematics vol 2 no 2pp 5ndash7 2012
[9] S Rashid N H Sarmin A Erfanian and N M Ali ldquoOn thenonabelian tensor square and capability of groups of order p2qrdquoArchiv der Mathematik vol 97 no 4 pp 299ndash306 2011
[10] S Rashid N H Sarmin A Erfanian N M Ali and R ZainalldquoOn the nonabelian tensor square and capability of groups oforder 8119902rdquo Indagationes Mathematicae New Series vol 24 no 3pp 581ndash588 2013
[11] M Quick MT5824 Topics in Groups Lecture Notes Universityof St Andrews 2004ndash2009
[12] M HallTheTheory of Groups Macmillan New York NY USA1959
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Case 2 If 119866119886119887 cong Z119901timesZ119901then by Proposition 11119872(119866) = Z
119901
(i) If 119873 = 1 then by Proposition 12 119872(119866119873) =
119872(119866 1) = 1(ii) If 119873 = 119866 then by Proposition 13 119872(119866119873) =
119872(119866 119866) = 119872(119866) By Proposition 11119872(119866) = Z119901
(iii) If119873 = Z119902then 119866 is the semidirect product ofZ
119902and
119867 in which |119873| and [119866 119873] are coprimes and 119873 isa normal Hall subgroup of 119866 (refer to Definition 1)Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since 119872(Z119902) = 1 (refer to Proposition 7) Note that
for this case 119866119873 = 1198661198661015840
= Z1199012 that is 119866119873 = Z
1199012
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 thatis Z119902sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If119873 = Z119901119902 Z1199012 or Z
119901timesZ119901then by similar way as in
(iv)119872(119866119873) = Z119901
Theorem 16 Let 119866 be a group of order 1199012119902 where 119901 and 119902 aredistinct primes and 119901 gt 119902 If119873 ⊲ 119866 then the Schur multiplierof pairs of 119866
119872(119866119873)
=
1 119894119891 1198661015840cong Z119901119900119903 1198661015840cong Z1199012 119900119903 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 1 119900119903 Z119902
Z119901 119894119891 119866
1015840cong Z119901times Z119901
119908ℎ119890119899 119873 = 119866 Z119901119900119903 Z119901times Z119901
(7)
where 119866119886119887 = 1198661198661015840
Proof Let119866be a groupof order1199012119902where119901 and 119902 are distinctprimes and 119901 gt 119902 Since 119901 gt 119902 119866 has a normal Sylow 119901-subgroup namely 119875 (refer to Proposition 3) [119866 119875] = 119902 so119866119875 is abelian Hence 1198661015840 sube 119875 (refer to Proposition 5) thatis 1198661015840 cong Z
119901times Z119901 Z1199012 or Z
119901 Suppose119873 ⊲ 119866 then the Schur
multiplier of pairs of 119866 is computed below(In this case119873 = 1Z
119902Z119901Z119901times Z119901and 119866)
Case 1 If 1198661015840 cong Z119901times Z119901then |119866
1015840| = 1199012 and [119866 119866
1015840] = 119902 are
coprimesThen byDefinition 11198661015840 is a normalHall subgroupof 119866 Therefore by Proposition 9 119872(119866) = 119872(119879) times 119872(119866
1015840)119879
where 119879 is a complement of 1198661015840 and 119879 cong Z119902 Thus 119872(119866) =
119872(119879) times 119872(Z119901times Z119901)119879 119872(119879) = 1 (refer to Proposition 7)
Hence119872(119866) = Z119901(refer to Propositions 10 6 and 7)
(i) If 119873 = 1 then 119872(119866119873) = 119872(119866 1) = 1 (refer toProposition 12)
(ii) If119873 = 119866 then119872(119866119873) = 119872(119866 119866) = 119872(119866) (refer toProposition 13) Then119872(119866) = Z
119901
(iii) If 119873 = Z119902then 119873 is a normal Hall subgroup of 119866
(refer to Definition 1) and119866 is the semidirect productof 119873 and 119867 in which 119867 is a complement of 119873 in 119866Therefore by Proposition 14119872(119866119873) = 119872(119873)
119867= 1
since119872(Z119902) = 1 (refer to Proposition 7)
(iv) If119873 = Z119901then119866119873 is nonabelian group of order 119901119902
(If 119866119873 cong Z119901119902
then by Proposition 5 1198661015840 sube 119873 that isZ119901times Z119901sube Z119901and this statement is a contradiction)
Thus the exact sequence 119872(119866119873) rarr 119872(119866) rarr
119872(119866119873) = 1 shows that 119872(119866119873)120581 cong Z119901where 120581
is the kernel of homomorphism 119872(119866119873) to 119872(119866)Then119872(119866119873) = Z
119901
(v) If 119873 = Z119901times Z119901then by similar way as in (iv)
119872(119866119873) = Z119901
(vi) If 119873 = Z119901119902
or Z1199012 then 119866119873 is abelian group and
1198661015840cong Z119901times Z119901sube 119873 cong Z
119901119902or Z1199012 but this statement
is a contradiction So119872(119866119873) when119873 = Z119901119902
orZ1199012
are not considered
Case 2 If 1198661015840 cong Z1199012 then 119866119866
1015840cong 119866119886119887
cong Z119902 Hence all
Sylow subgroups of 119866 are cyclic Therefore by Proposition 8119872(119866) = 1Thus for all normal subgroups119873 of119866119872(119866119873) le
119872(119866) = 1
Case 3 If 1198661015840 cong Z119901then 119872(119866) = 1 since 119872(119866) = 119872(119866
1015840) times
119872(119870) where119870 is a group of order 119901119902 By Propositions 7 and8119872(119866
1015840) = 1 and119872(119870) = 1 Thus for all normal subgroups
119873 of 119866119872(119866119873) le 119872(119866) = 1
4 Conclusion
For a group119866 of order 1199012119902where 119901 and 119902 are prime numbers119876 is the unique normal Sylow 119902-subgroups of 119866 if 119901 lt 119902while 119875 is the unique normal Sylow 119901-subgroups of 119866 if 119901 gt
119902 In this paper we determined the Schur multiplier of pairsof groups of order 1199012119902 Our proofs show that 119872(119866119873) forgroups of order 1199012119902 is either 1 or Z
119901
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank Ministry of Education(MOE)Malaysia and ResearchManagement Centre Univer-siti TeknologiMalaysia (RMCUTM) for the financial supportthrough the Research University Grant (RUG) Vote no04H13 The second author would also like to thank Ministryof Education (MOE) Malaysia for her MyPhD Scholarship
References
[1] I Schur ldquoUber die Darstellung der endlichen gruppen durchgebrochen lineare substitutionenrdquo Journal fur die Reine und
4 Mathematical Problems in Engineering
Angewandte Mathematik vol 1904 no 127 pp 20ndash50 19041904
[2] G KarpilovskyThe Schur Multiplier Clarendon Press OxfordUK 1987
[3] G Ellis ldquoThe Schur multiplier of a pair of groupsrdquo AppliedCategorical Structures vol 6 no 3 pp 355ndash371 1998
[4] R Brown and J-L Loday ldquoVan Kampen theorems for diagramsof spacesrdquo Topology vol 26 no 3 pp 311ndash335 1987
[5] M Visscher On the nonabelian tensor products of groups [PhDdissertation] State University of NewYork at Binghamton 1998
[6] S H Jafari P Niroomand and A Erfanian ldquoThe nonabeliantensor square and Schur multiplier of groups of order 1199012119902 1199011199022and 119901
2119902119903rdquo Algebra Colloquium vol 10 pp 1083ndash1088 2012
[7] M R R Moghaddam A R Salemkar and K Chiti ldquoSomeproperties on the Schur multiplier of a pair of groupsrdquo Journalof Algebra vol 312 no 1 pp 1ndash8 2007
[8] S Rashid N H Sarmin A Erfanian and N M Mohd Ali ldquoOnthe commutator subgroups of groups of order 8qrdquo Journal ofComputer Science amp Computational Mathematics vol 2 no 2pp 5ndash7 2012
[9] S Rashid N H Sarmin A Erfanian and N M Ali ldquoOn thenonabelian tensor square and capability of groups of order p2qrdquoArchiv der Mathematik vol 97 no 4 pp 299ndash306 2011
[10] S Rashid N H Sarmin A Erfanian N M Ali and R ZainalldquoOn the nonabelian tensor square and capability of groups oforder 8119902rdquo Indagationes Mathematicae New Series vol 24 no 3pp 581ndash588 2013
[11] M Quick MT5824 Topics in Groups Lecture Notes Universityof St Andrews 2004ndash2009
[12] M HallTheTheory of Groups Macmillan New York NY USA1959
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Angewandte Mathematik vol 1904 no 127 pp 20ndash50 19041904
[2] G KarpilovskyThe Schur Multiplier Clarendon Press OxfordUK 1987
[3] G Ellis ldquoThe Schur multiplier of a pair of groupsrdquo AppliedCategorical Structures vol 6 no 3 pp 355ndash371 1998
[4] R Brown and J-L Loday ldquoVan Kampen theorems for diagramsof spacesrdquo Topology vol 26 no 3 pp 311ndash335 1987
[5] M Visscher On the nonabelian tensor products of groups [PhDdissertation] State University of NewYork at Binghamton 1998
[6] S H Jafari P Niroomand and A Erfanian ldquoThe nonabeliantensor square and Schur multiplier of groups of order 1199012119902 1199011199022and 119901
2119902119903rdquo Algebra Colloquium vol 10 pp 1083ndash1088 2012
[7] M R R Moghaddam A R Salemkar and K Chiti ldquoSomeproperties on the Schur multiplier of a pair of groupsrdquo Journalof Algebra vol 312 no 1 pp 1ndash8 2007
[8] S Rashid N H Sarmin A Erfanian and N M Mohd Ali ldquoOnthe commutator subgroups of groups of order 8qrdquo Journal ofComputer Science amp Computational Mathematics vol 2 no 2pp 5ndash7 2012
[9] S Rashid N H Sarmin A Erfanian and N M Ali ldquoOn thenonabelian tensor square and capability of groups of order p2qrdquoArchiv der Mathematik vol 97 no 4 pp 299ndash306 2011
[10] S Rashid N H Sarmin A Erfanian N M Ali and R ZainalldquoOn the nonabelian tensor square and capability of groups oforder 8119902rdquo Indagationes Mathematicae New Series vol 24 no 3pp 581ndash588 2013
[11] M Quick MT5824 Topics in Groups Lecture Notes Universityof St Andrews 2004ndash2009
[12] M HallTheTheory of Groups Macmillan New York NY USA1959
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
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