# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

##### 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
Full Text:
##### References:
 [1] Aboughazi, R.: Produit tensoriel du groupe d’heisenberg. Bull. soc. Math. France 115, 95-106 (1987) · Zbl 0619.57001 [2] Beyl, R.: The Schur multiplicator of metacyclic groups. Proc. amer. Math. soc. 40, 413-418 (1973) · Zbl 0267.18021 [3] Brown, R.; Loday, J. -L: Excision homotopique en basse dimension. C. R. Acad. sci. Paris sér. I math. 298, No. No. 15, 353-356 (1984) · Zbl 0573.55011 [4] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology, in press. · Zbl 0622.55009 [5] Coxeter, H. S. M; Moser, W. O. J: Generators and relations for discrete groups. (1980) · Zbl 0422.20001 [6] Dennis, R. K.: In search of new ”homology” functors having a close relationship to K-theory. (1976) [7] Ellis, G. J.: Crossed modules and their higher dimensional analogues. Ph.d. thesis (1984) [8] Ellis, G. J.: Non-abelian exterior products of groups and exact sequences in the homology of groups. Glasgow math. J. 29, 13-19 (1987) · Zbl 0631.20040 [9] G. J. Ellis, The non-abelian tensor product of finite groups is finite, J. Algebra, in press. · Zbl 0626.20039 [10] Guin, D.: Cohomologie et homologie non-abélienne des groupes. (1985) · Zbl 0584.18006 [11] Havas, G.; Kenne, P. E.; Richardson, J. S.; Robertson, E. F.: A tietze transformation program. Computational group theory, 69-73 (1984) · Zbl 0569.20002 [12] Huppert, B.: Endliche gruppen, I. (1967) [13] Maclane, S.: Homology. (1963) · Zbl 0133.26502 [14] Miller, C.: The second homology of a group. Proc. amer. Math. soc. 3, 588-595 (1952) · Zbl 0047.25703 [15] Whitehead, J. H. C: A certain exact sequence. Ann. of math. 52, 51-110 (1950) · Zbl 0037.26101 [16] Gilbert, N. D.: The non-abelian tensor square of a free product of groups. Arch. math. 48, 369-375 (1987) · Zbl 0635.20012 [17] N. D. Gilbert and P. J. Higgins, The non-abelian tensor product of groups and related constructions, U.C.N.W. Pure Maths, Preprint 87.11. · Zbl 0681.20034 [18] Johnson, D. L.: The non-abelian tensor square of a finite split metacyclic group. Proc. Edinburgh math. Soc. 30, 91-96 (1987) · Zbl 0588.20022 [19] Lue, A. S. -T: The ganea map for nilpotent groups. J. London math. Soc. 14, 309-312 (1976) · Zbl 0357.20030