A combinatorial problem in infinite groups.

zbMATH Open1185.20030arXivmath/0212017MaRDI QIDQ968660

Alireza Abdollahi

Publication date: 5 May 2010

Published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series (Search for Journal in Brave)

Abstract: Let $w$ be a word in the free group of rank $ninmathbbN$ and let $mathcalV(w)$ be the variety of groups defined by the law $w=1$. Define $mathcalV(w^{*})$ to be the class of all groups $G$ in which for any infinite subsets $X_1,...,X_n$ there exist $x_{i}in{X}_i$, $1leqileqn$, such that $w(x_1,...,x_n)=1$. Clearly, $mathcalV(w)cupmathcalFsubseteqmathcalV(w^{*})$; $mathcalF$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $mathcalP$ of groups for which the equality $(mathcalV(w)cupmathcalF)capmathcalP=mathcalPcapmathcalV(w^{*})$ holds.

Full work available at URL: https://arxiv.org/abs/math/0212017

zbMATH Keywords

varieties of groupsfree groupscombinatorial conditions on infinite subsetslaws

Mathematics Subject Classification ID

Special aspects of infinite or finite groups (20F99) Quasivarieties and varieties of groups (20E10)

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