Some Inequalities for Nilpotent Multipliers of Powerful $p$-Groups

TL;DR

This paper establishes new inequalities relating to the order, exponent, and generators of the c-nilpotent multiplier in powerful p-groups, extending previous results and providing explicit examples for validation.

Contribution

It introduces novel inequalities for the c-nilpotent multiplier of powerful p-groups, extending prior work and including explicit examples to demonstrate their tightness.

Findings

Derived inequalities for order, exponent, and generators of the c-nilpotent multiplier.

Extended Lubotzky and Mann's results to nilpotent multipliers.

Provided explicit examples confirming the sharpness of the inequalities.

Abstract

In this paper we present some inequalities for the order, the exponent, and the number of generators of the c-nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c \geq 1$) of a powerful p-group. Our results extend some of Lubotzky and Mann$^{,}$s (Journal of Algebra, 105 (1987), 484-505.) to nilpotent multipliers. Also, we give some explicit examples showing the tightness of our results and improvement some of the previous inequalities.