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The commutator subgroup and Schur multiplier of a pair of finite p-groups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Ali Reza Salemkar ,
Mohammad Reza R. Moghaddam  and
Farshid Saeedi
Ali Reza Salemkar
Affiliation:
Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran, e-mail: salemkar@usb.ac.ir
Mohammad Reza R. Moghaddam
Affiliation:
Centre of Excelence in Analysis on Algebraic Structures (President), and Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran, e-mail: moghadam@math.um.ac.ir
Farshid Saeedi
Affiliation:
Department of Mathematics, Azad University of Mashhad, Iran, e-mail: saeedi@mshdiau.ac.ir
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Abstract

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Let (M, G) be a pair of groups, in which M is a normal subgroup of G such that G/M and M/Z(M, G) are of orders pm and pn. respectively. In 1998, Ellis proved that the commutator subgroup [M, G] has order at most pn(n + 2 m−1)/2.

In the present paper by assuming /[M, G] = pn(n+2m−1)/2, we determine the pair (M, G). An upper bound is obtained for the Schur multiplier of the pair (M, G), which generalizes the work of Green (1956).

Keywords

pair of groupsSchur multiplierrelative central extensioncovering pairextra-special pair.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups 20E34: General structure theorems 20E36: Automorphisms of infinite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 1 , August 2006 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Society 2006

References

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