*(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

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 $G=\langle x,y\mid {x}^{m}=e={y}^{n}$, $xy{x}^{-1}={y}^{\ell}\rangle $, where ${\ell}^{m}=1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$ 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.

##### 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 |