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On Peiffer central series

Published online by Cambridge University Press:  18 May 2009

Graham Ellis
Graham Ellis
Affiliation:
Department of Mathematics, University College Galway, Ireland E-mail: graham.ellis@ucg.ie
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Let G be a group. A precrossed G-module is a group homomorphism ∂: M → G together with a group action (g, m) ↦gm of G on M, such that ∂(gm) = g(∂m)g−1. The Peiffer commutator < m, m′ > of two elements m, m′ ∊ M is denned as

< m, m′ >= mm′ m−1(∂mm′)−1

If all Peiffer commutators are trivial, the precrossed G-module is said to be a crossed G-module. The subgroup < M, M > generated by all Peiffer commutators is called the Peiffer subgroup of M; it is the second term of a lower Peiffer central series (see below). The following table indicates how these concepts reduce to more standard concepts when restrictions are placed on ∂ and G.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 40 , Issue 2 , May 1998 , pp. 177 - 185
Copyright
Copyright © Glasgow Mathematical Journal Trust 1998

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