Finite index and finite codimension
type: Item
instance of: scholarly article
title: Finite index and finite codimension
zbMATH Open document ID: 0842.20024
DOI: 10.1016/0022-4049(94)00120-8
author: Shmuel Rosset
published in: Journal of Pure and Applied Algebra
publication date: 1996-07-22T00:00:00Z
review text: Let \(G\) be a finitely generated free group. If \(H\) is a finitely generated subgroup of \(G\), then it is shown that there exists a subgroup \(\text{root}_G(H)\), the root of \(H\) in \(G\), containing \(H\), in which \(H\) has finite index and which contains every subgroup with this property. The proof is based on a theorem of \textit{M. Hall} [Trans. Am. Math. Soc. 67, 421-432 (1949; Zbl 0035.01301)] which asserts the existence of complements \(K\) with the property that the natural map \(H*K\to H\vee K\), the subgroup generated by \(H\) and \(K\), is an isomorphism and the index \((G:H\vee K)\) is finite. Analogous theory of roots of ideals in free algebra \(k\langle X\rangle\) and free group ring \(k(G)\) over a field \(k\) is developed. The author defines the infimum, when \(K\) runs over all Hall complements of \(H\), to be the Hall index \((G:H)_{\text{Hall}}\) of \(H\) in \(G\) and proves that if \(A\) and \(B\) are finitely generated subgroups of \(G\), then \[ \text{rank}(A\cap B)\leq 1+\min\{(G:A)_{\text{Hall}}\cdot(\text{rank}(B)-1),(G:B)_{\text{Hall}}\cdot(\text{rank}(A)-1)\}. \]
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 825328
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: roots of subgroups
zbMATH Keywords: finitely generated free groups
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: existence of complements
zbMATH Keywords: roots of ideals
zbMATH Keywords: free algebras
zbMATH Keywords: free group rings
zbMATH Keywords: Hall complements
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00120-8
OpenAlex ID: W2013923794
cites work: Coset Representations in Free Groups
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cites work: Q4145882
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cites work: On equivalent sets of elements in a free group
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description: scientific article; zbMATH DE number 825328