Criteria for laws between infinite subsets of infinite groups

Abstract

Year: 2010 Vol.: 77 Fasc.: 3-4 Title: Criteria for laws between in¯nite subsets of in¯nite groups Author(s): A. Boukaroura and M. Bouchlaghem A theorem of B. H. Neumman shows that in¯nite group in which every two in¯nite subsets there exist two commuting elements, is abelian. In this paper, we prove that if in an in¯nite group G, every two in¯nite subsets X and Y , there exist a 2 X and b 2 Y such that [an1 ; bn2 ] = 1, then G satis¯es the law [xn1 ; yn2 ] = 1, where n2 ´ 0[n1] and n2 2 f3; 6; 2k=k 2 N¤g. Moreover, and using this result, we also prove that an in¯nite group satis¯es the law (xn1 1 xn2 2 : : : xnr r )2 = 1 if and only if in any r in¯nite subsets X1; : : : ;Xr, of G there exist ai 2 Xi(i = 1; : : : ; r) such that (an1 1 : : : anr r )2 = 1, where n1; : : : ; nr 2 f2k=k 2 N¤g and r ¸ 2. Address: A. Boukaroura L.M.F.N, Department of Mathematics University of Ferhat Abbas Setif 19000 Algeria Address: M. Bouchlaghem L.M.F.N, Department of Mathematics University of Ferhat Abbas Setif 19000 Algeria