# Classification of finite $p$-groups by the size of their Schur multipliers

## TL;DR

This paper characterizes the structure of finite p-groups that reach the maximum possible order of their Schur multiplier, based on their minimal generating set and derived subgroup size, and shows these groups are capable.

## Contribution

It determines the structure of all p-groups attaining the upper bound of their Schur multiplier's order and proves their capability.

## Key findings

- Identified the structure of p-groups reaching the Schur multiplier bound.
- Proved all such groups are capable.
- Provided a classification for these extremal p-groups.

## Abstract

Let $d(G)$ be the minimum number of elements required to generated a group $G.$ For a group $G $ of order $p^n$ with derived subgroup of order $ p^k $ and $d(G) = d,$ we knew the order of the Schur multiplier of $G$ is bounded by $ p^{\frac{1}{2}(d-1)(n-k+2)+1}. $ In the current paper, we find the structure of all $p$-groups that attains the mentioned bound. Moreover, we show that all of them are capable.