On the Order of Nilpotent Multipliers of Finite p-Groups

TL;DR

This paper investigates the upper bounds and structure of the $c$-nilpotent multipliers of finite $p$-groups, establishing conditions under which these groups are elementary abelian based on the order of their multipliers.

Contribution

It derives an explicit upper bound for the order of the $c$-nilpotent multiplier of finite $p$-groups and characterizes the structure of groups attaining this bound.

Findings

Upper bound $p^{inom{n}{c+1}}$ for the order of $c$-nilpotent multiplier

Structure characterization of abelian groups with maximum $c$-nilpotent multiplier

Elementary abelian $p$-groups are characterized by maximum $c$-nilpotent multiplier order

Abstract

Let $G$ be a finite $p$-group of order $p^n$. YA. G. Berkovich (Journal of Algebra {\bf 144}, 269-272 (1991)) proved that $G$ is elementary abelian $p$-group if and only if the order of its Schur multiplier, $M(G)$, is at the maximum case. In this paper, first we find the upper bound $p^{\chi_{c+1}{(n)}}$ for the order the $c$-nilpotent multiplier of $G$, $M^{(c)}(G)$, where $\chi_{c+1}{(i)}$ is the number of basic commutators of weight $c+1$ on $i$ letters. Second, we obtain the structure of $G$, in abelian case, where $|M^{(c)}(G)|=p^{\chi_{c+1}{(n-t)}}$, for all $0\leq t\leq n-1$. Finally, by putting a condition on the kernel of the left natural map of the generalized Stallings-Stammbach five term exact sequence, we show that an arbitrary finite $p$-group with the $c$-nilpotent multiplier of maximum order is an elementary abelian $p$-group.