The Schur multiplier and capability of pairs of some finite groups

The homological functors of a group have its origin in homotopy theory and algebraic K-theory. The Schur multiplier of a group is one of the homological functors while the Schur multiplier of pairs of groups is an extension of the Schur multiplier of a group. Besides, a pair of groups is capable if...

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Main Author:	Nawi, Adnin Afifi
Format:	Thesis
Language:	English
Published:	2017
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/81764/1/AdninAfifiNawiPFS2017.pdf
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spelling	my-utm-ep.817642019-09-29T10:53:47Z The Schur multiplier and capability of pairs of some finite groups 2017-06 Nawi, Adnin Afifi QA Mathematics The homological functors of a group have its origin in homotopy theory and algebraic K-theory. The Schur multiplier of a group is one of the homological functors while the Schur multiplier of pairs of groups is an extension of the Schur multiplier of a group. Besides, a pair of groups is capable if the precise center or epicenter of the pair of groups is trivial. In this research, the Schur multiplier and capability of pairs of groups of order p, p2, p3, p4, pq, p2q and p3q (where p and q are distinct odd primes) were determined. This research started with the computation of the normal subgroups of the groups. The generalized structures of the normal subgroups have been found by the assistance of Groups, Algorithms and Programming (GAP) software. By using the Sylow theorems and the results of the nonabelian tensor products, derived subgroups, centers of the groups, abelianization of the groups and the Schur multiplier of the groups, the Schur multiplier of pairs of groups of order p, p2, pq, p2q, p3, p4 and p3q were then determined. The classification of the groups had also been used in the computation of the Schur multiplier and capability of pairs of groups. The order of the epicenter of pairs of groups of order p, p2, p3, p4 and pq were also computed by using GAP software to determine the capability of pairs of the groups. The Schur multiplier of pairs of group is found to be trivial or abelian. All pairs of groups where G is isomorphic to the elementary abelian groups of order p2, p3 and p4, nonabelian group of order p3 of exponent p, direct product of two cyclic groups of order p2, semi- direct product of two cyclic groups of order p2, and direct product of cyclic group of order p and nonabelian group of order p3 of exponent p are capable. For other groups, only certain pairs of groups are capable depending on their normal subgroups. 2017-06 Thesis http://eprints.utm.my/id/eprint/81764/ http://eprints.utm.my/id/eprint/81764/1/AdninAfifiNawiPFS2017.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:126138 phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Nawi, Adnin Afifi The Schur multiplier and capability of pairs of some finite groups
description	The homological functors of a group have its origin in homotopy theory and algebraic K-theory. The Schur multiplier of a group is one of the homological functors while the Schur multiplier of pairs of groups is an extension of the Schur multiplier of a group. Besides, a pair of groups is capable if the precise center or epicenter of the pair of groups is trivial. In this research, the Schur multiplier and capability of pairs of groups of order p, p2, p3, p4, pq, p2q and p3q (where p and q are distinct odd primes) were determined. This research started with the computation of the normal subgroups of the groups. The generalized structures of the normal subgroups have been found by the assistance of Groups, Algorithms and Programming (GAP) software. By using the Sylow theorems and the results of the nonabelian tensor products, derived subgroups, centers of the groups, abelianization of the groups and the Schur multiplier of the groups, the Schur multiplier of pairs of groups of order p, p2, pq, p2q, p3, p4 and p3q were then determined. The classification of the groups had also been used in the computation of the Schur multiplier and capability of pairs of groups. The order of the epicenter of pairs of groups of order p, p2, p3, p4 and pq were also computed by using GAP software to determine the capability of pairs of the groups. The Schur multiplier of pairs of group is found to be trivial or abelian. All pairs of groups where G is isomorphic to the elementary abelian groups of order p2, p3 and p4, nonabelian group of order p3 of exponent p, direct product of two cyclic groups of order p2, semi- direct product of two cyclic groups of order p2, and direct product of cyclic group of order p and nonabelian group of order p3 of exponent p are capable. For other groups, only certain pairs of groups are capable depending on their normal subgroups.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Nawi, Adnin Afifi
author_facet	Nawi, Adnin Afifi
author_sort	Nawi, Adnin Afifi
title	The Schur multiplier and capability of pairs of some finite groups
title_short	The Schur multiplier and capability of pairs of some finite groups
title_full	The Schur multiplier and capability of pairs of some finite groups
title_fullStr	The Schur multiplier and capability of pairs of some finite groups
title_full_unstemmed	The Schur multiplier and capability of pairs of some finite groups
title_sort	schur multiplier and capability of pairs of some finite groups
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2017
url	http://eprints.utm.my/id/eprint/81764/1/AdninAfifiNawiPFS2017.pdf
_version_	1747818408365260800

The Schur multiplier and capability of pairs of some finite groups

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