Tame almost primitive elements
type: Item
instance of: scholarly article
title: Tame almost primitive elements
zbMATH Open document ID: 0962.20022
DOI: 10.1007/BF03322435
author: Jochen Konieczny
author: Gerhard Rosenberger
author: Julia Wolny
published in: Results in Mathematics
publication date: 2000-08-31T00:00:00Z
review text: Let \(G\) be a finitely generated group, \(V\) a set of laws, and \(\mathcal V\) the variety defined by \(V\). An element \(\omega\in G\) is \(\mathcal V\)-generic in \(G\) if \(\omega\in V(G)\) but \(\omega\notin V(K)\) for any proper subgroup \(K<G\), primitive if it is contained in some minimal generating set for \(G\), almost primitive if it is primitive in every proper subgroup of \(G\) which contains it, and tamely almost primitive if it is almost primitive, and whenever any power \(\omega^\alpha\in H\leq G\) where \(\alpha\) is positive and minimal and \(H\) is finitely-generated, then either \(\omega^\alpha\) is primitive in \(H\) or the index of \(H\) in \(G\) is \(\alpha\). The first result of the paper shows that \(a^\alpha b^\beta\) is tame almost primitive in the free group \(F_2=\langle a,b\rangle\) if and only if \(\alpha=\beta=2\). The remaining sections of the paper are devoted to the study of generic, almost primitive and tame almost primitive elements in surface groups. Let \(G=\langle a_1,\dots,a_n;\;[a_1,a_2]\cdots[a_{n-1},a_n]=1\rangle\), \(n\geq 4\), \(n\) even. It is shown that \(\omega=a_1^{p_1}\cdots a_n^{p_n}\), where all the \(p_i\) are primes, is almost primitive, but it is tamely almost primitive if and only if all the \(p_i\) are equal to 2.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 1498582
zbMATH Keywords: tame almost primitive elements
zbMATH Keywords: generic elements
zbMATH Keywords: finitely generated groups
zbMATH Keywords: minimal generating sets
zbMATH Keywords: free groups
zbMATH Keywords: varieties of groups
zbMATH Keywords: surface groups
reviewed by: Sheila Oates-Williams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1498582