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Some computations of non-Abelian tensor products of groups. (English) Zbl 0626.20038

Let G and H be groups which act on themselves by conjugation and with a compatible action of G on H and of H on G. Then the non-Abelian tensor product GH is the group generated by the symbols gh subject to the relations

gg ' h=( g g ' g h)(gh),ghh ' =(gh)( h g h h ' )forallg,g ' Gandh,h ' H·

The authors in the present paper are mainly concerned with the computation of GG. Let A,B,C be groups with given actions of A on B and C and of B and C on A. Under suitable conditions on these actions it is proved that A(BC)ABAC. The tensor squares GG when G is (i) the quaternion group of order 4m; (ii) the dihedral group of order 2m; (iii) the metacyclic group G=x,yx m =e=y n , xyx -1 =y , where m =1(modn) and n is odd; are computed. Another interesting result proved is that GG is the unique covering group of the perfect group G. The tensor squares GG for non-Abelian groups of order 30 obtained by using the Tietze transformation program are given. Also given are the generators and relations for GG for some of these groups. Some open problems are listed.

Reviewer: L.R.Vermani

20J05Homological methods in group theory
20J06Cohomology of groups
20E22Extensions, wreath products, and other compositions of groups
20F05Generators, relations, and presentations of groups