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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 $$gg'\otimes h=(\sp gg'\otimes\sp gh)(g\otimes h),\quad g\otimes hh'=(g\otimes h)(\sp hg\otimes\sp hh')\text{ for all } g,g'\in G\text{ and } h,h'\in H.$$ 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 $A\otimes (B\oplus C)\cong A\otimes B\oplus A\otimes C$. 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 $G=\langle x,y\mid x\sp m=e=y\sp n$, $xyx\sp{-1}=y\sp{\ell}\rangle$, where $\ell\sp m=1\pmod n$ 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

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