A combinatorial condition on a certain variety of groups |
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Authors: | B Taeri |
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Institution: | Department of Mathematics, Isfahan University of Technology, Isfahan, Iran, b.taeri@cc.iut.ac.ir, IR
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Abstract: | Let n be an integer and Bn \mathcal B_n be the variety defined by the law xn,y]x,yn]-1 = 1.¶ Let Bn* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that xn,y]x,yn]-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ Bn* = Bn èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? Bn* G \in \mathcal B_n^* if and only if G ? Bn G \in \mathcal B_n . |
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