Triple exterior products and 2-nilpotent multipliers of finite split metacyclic groups

Abstract

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group G be presented as the quotient of a free group F by a normal subgroup R. Given a positive integer c, the c-nilpotent multiplier of the group G is the abelian group ${\mathcal{M}}^{(c)}(G)=(R\cap {\gamma }_{c+1}(F))/{\gamma }_{c+1}(R,F)$. In particular, ${\mathcal{M}}^{(1)}(G)$ is the Schur multiplier of G. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups.

Keywords:

• Group action
• metacyclic group
• nilpotent multiplier
• nonabelian tensor product

2020 Mathematics Subject Classification:

• 20F05
• 20C25
• 20J99

Acknowledgments

The authors are very grateful to the referee for many insightful comments and helpful suggestions on an earlier version of this paper.

Disclosure statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.