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

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.

arXiv commentSome results and proofs are modified in the current version, typos are corrected and the title of the paper is changed according to the content