Embedding into k-efficient groups

Graham Ellis

doi:10.1006/jabr.2001.8811

Embedding into k-efficient groups

Graham Ellis

Mathematics

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Abstract

We prove a theorem with the following corollary: For each integer k ≥ 1, an arbitrary finite group G embeds into some finite group G_k for which there exists an Eilenberg-Mac Lane CW-space X = K(G_k, 1) whose finite n-skeleton Xⁿ has Euler-Poincaré characteristics _X(Xⁿ) = 1 + (-1)ⁿdH_n(G_k) for all n ≤ k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra 182, 511-521] on the embedding of finite groups into groups with "efficient" presentations.

Original language	English
Pages (from-to)	497-503
Number of pages	7
Journal	Journal of Algebra
Volume	243
Issue number	2
DOIs	https://doi.org/10.1006/jabr.2001.8811
Publication status	Published - 15 Sep 2001

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title = "Embedding into k-efficient groups",

abstract = "We prove a theorem with the following corollary: For each integer k ≥ 1, an arbitrary finite group G embeds into some finite group Gk for which there exists an Eilenberg-Mac Lane CW-space X = K(Gk, 1) whose finite n-skeleton Xn has Euler-Poincar{\'e} characteristics X(Xn) = 1 + (-1)ndHn(Gk) for all n ≤ k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra 182, 511-521] on the embedding of finite groups into groups with {"}efficient{"} presentations.",

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N2 - We prove a theorem with the following corollary: For each integer k ≥ 1, an arbitrary finite group G embeds into some finite group Gk for which there exists an Eilenberg-Mac Lane CW-space X = K(Gk, 1) whose finite n-skeleton Xn has Euler-Poincaré characteristics X(Xn) = 1 + (-1)ndHn(Gk) for all n ≤ k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra 182, 511-521] on the embedding of finite groups into groups with "efficient" presentations.

AB - We prove a theorem with the following corollary: For each integer k ≥ 1, an arbitrary finite group G embeds into some finite group Gk for which there exists an Eilenberg-Mac Lane CW-space X = K(Gk, 1) whose finite n-skeleton Xn has Euler-Poincaré characteristics X(Xn) = 1 + (-1)ndHn(Gk) for all n ≤ k. The theorem can be viewed as a generalisation of a result of J. Harlander [1996, J. Algebra 182, 511-521] on the embedding of finite groups into groups with "efficient" presentations.

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DO - 10.1006/jabr.2001.8811

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VL - 243

SP - 497

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JO - Journal of Algebra

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IS - 2

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Embedding into k-efficient groups

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