ON NONABELIAN TENSOR PRODUCT MODULO q OF GROUPS

Abstract

The nonabelian tensor product modulo q of two crossed modules of groups is investigated, where q is a positive integer. It is obtained a six term exact sequence of groups connecting the nonabelian tensor product modulo q with algebraic K-functor K ₂ with Z _q coefficients for (noncommutative) local rings. The notion of q-homology groups of a group G with coefficients in a G-module A is introduced, some its properties and calculations are given. The relationship between q-homology groups and derived functors of tensor product modulo q is studied.

Keywords:

• Tensor product modulo q
• K-functor
• q-homology of groups
• Derived functors
• 1991 Mathematics Subject Classification: 20J05, 18G10, 18G50, 19C20

ACKNOWLEDGMENTS

The author would like to thank his father Prof. H. Inassaridze for discussions throughout the period of the work on this paper and Prof. G. Ellis and Prof. D. Conduche for helpful comments. The work was partially supported by Grant No. GM1-115 of the U.S. CRDF, INTAS Georgia grant No. 213 and INTAS Fellowship grant for Young Scientists No. 151.