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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 $G\otimes H$ is the group generated by the symbols $g\otimes h$ subject to the relations

$g{g}^{\text{'}}\otimes h={\left(}^{g}{g}^{\text{'}}{\otimes }^{g}h\right)\left(g\otimes h\right),\phantom{\rule{1.em}{0ex}}g\otimes h{h}^{\text{'}}=\left(g\otimes h\right){\left(}^{h}g{\otimes }^{h}{h}^{\text{'}}\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}g,{g}^{\text{'}}\in G\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}h,{h}^{\text{'}}\in H·$

The authors in the present paper are mainly concerned with the computation of $G\otimes G$. 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 $A\otimes \left(B\oplus C\right)\cong A\otimes B\oplus A\otimes C$. The tensor squares $G\otimes G$ when $G$ is (i) the quaternion group of order $4m$; (ii) the dihedral group of order $2m$; (iii) the metacyclic group $G=〈x,y\mid {x}^{m}=e={y}^{n}$, $xy{x}^{-1}={y}^{\ell }〉$, where ${\ell }^{m}=1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}n\right)$ and $n$ is odd; are computed. Another interesting result proved is that $G\otimes G$ is the unique covering group of the perfect group $G$. The tensor squares $G\otimes G$ for non-Abelian groups of order $\le 30$ obtained by using the Tietze transformation program are given. Also given are the generators and relations for $G\otimes G$ for some of these groups. Some open problems are listed.

Reviewer: L.R.Vermani

##### MSC:
 20J05 Homological methods in group theory 20J06 Cohomology of groups 20E22 Extensions, wreath products, and other compositions of groups 20F05 Generators, relations, and presentations of groups