The schur multipliers, nonbelian tensor squares and capability of some finite p-groups

Abstract

The homological functors and nonabelian tensor product have its roots in algebraic K-theory as well as in homotopy theory. Two of the homological functors are the Schur multiplier and nonabelian tensor square, where the nonabelian tensor square is a special case of the nonabelian tensor product. A group is said to be capable if it is a central factor group. In this research, the Schur multiplier, nonabelian tensor square and capability for some groups of order p3, p4, p5 and p6 are determined. An algebraic computation of the center, derived subgroups, abelianization, Schur multipliers, nonabelian tensor squares and capability of the groups are determined with the assistance of Groups, Algorithms and Programming (GAP) software. Using the results of the center, derived subgroups and abelianization, the Schur multiplier, nonabelian tensor square and capability for the groups are determined. The nonabelian tensor squares and capability are also determined using the results of the Schur multipliers. The Schur multiplier of each of the groups considered is found to be trivial or abelian. The results show that the nonabelian tensor square of the groups are always abelian. In addition, a group has been shown to be capable if it has a nontrivial kernel or it is an extra-special p-group with exponent p.