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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 \[ gg'\otimes h=(^ gg'\otimes^ gh)(g\otimes h),\quad g\otimes hh'=(g\otimes h)(^ hg\otimes^ hh')\text{ for all } g,g'\in G\text{ and } h,h'\in H. \] 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 \(G=\langle x,y\mid x^ m=e=y^ n\), \(xyx^{-1}=y^{\ell}\rangle\), where \(\ell^ m=1\pmod n\) 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 \(\leq 30\) 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
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References:

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