On the Order of Schur Multipliers of Finite Abelian p-Groups

TL;DR

This paper investigates the structure of finite abelian p-groups based on the order of their Schur multipliers, extending previous classifications for small values of a parameter t.

Contribution

It introduces new structural results for abelian p-groups considering the exponents of the group, its Schur multiplier, and the metabelian multiplier.

Findings

Characterization of abelian p-groups with specific Schur multiplier orders

Structural conditions based on exponents of G, M(G), and S_2M(G)

Extension of known classifications for t=0,1,2,3

Abstract

Let $G$ be a finite $p$-group of order $p^{n}$ with $|M(G)|=p^{\frac{n(n-1)}{2}-t},$ where $M(G)$ is the Schur multiplier of $G$. Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of $G$ when $t=0,1,2,3$. In this paper, we are going to find some structures for an abelian $p$-group $G$ with conditions on the exponents of $G, M(G),$ and $S_2M(G)$, where $S_2M(G)$ is the metabelian multiplier of $G$.