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A Combinatorial Problem in Infinite Groups

Abdollahi, A., (2002) A Combinatorial Problem in Infinite Groups. Bulletin of the Malaysian Mathematical Sciences Society, 25 (2). pp. 101-114. ISSN 0126-6705

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Official URL: http://math.usm.my/bulletin/pdf/v25n2/v25n2p2.pdf

Affiliations

University of Isfahan, Iran, Dept. of Mathematics

Abstract

Let $w$ be a word in the free group of rank $n \in IN$ and let $V(w)$ be the variety of groups defined by the law $w = 1$. Define $V(w^\ast )$ to be the class of all groups $G$ in which for any $n$ infinite subsets $X_1,\cdots ,X_n$ there exist $X_i \in X_i , 1 \leq i \leq n$, such that $w(x_1,\cdots ,x_n ) = 1$. Clearly, $V(w) \cup \mathcal{F} \subseteq V(w^\ast ); \mathcal{F}$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $ \mathcal{P}$ of groups for which the equality $(V(w) \cup \mathcal{F}) \cap \mathcal{P} = V(w^\ast ) \cap \mathcal{P} $ holds.

Item Type:Journal
Subjects:Q Science, Computer Science
ID Code:1352

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