Bounds On the order of the Schur multiplier of $p$-groups

TL;DR

This paper refines bounds on the order of the Schur multiplier for p-groups by incorporating the group's nilpotency class, providing tighter estimates especially for classes 2 and maximal class.

Contribution

It introduces a new bound on the Schur multiplier of p-groups that includes the nilpotency class as a parameter, improving previous bounds.

Findings

Derived tighter bounds for nilpotency class 2 p-groups

Established bounds for maximal class p-groups

Enhanced understanding of Schur multiplier behavior in p-groups

Abstract

In 1956, Green provided a bound on the order of the Schur multiplier of $p$-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other inputs to the function, such as, the minimal number of generators of the group and the order of the derived subgroup. We strengthen these bounds by adding another input, the group's nilpotency class. The specific cases of nilpotency class 2 and maximal class are discussed in greater detail.