A lemma on the exponent of Schur multiplier of $p$ groups with good power structure

TL;DR

This paper provides concise proofs that the exponent of the Schur multiplier divides the exponent of certain finite p-groups with specific power structures, extending known results to new classes of groups.

Contribution

It introduces a general lemma and applies it to prove the exponent divisibility for maximal class, potent, and specific 3-groups, broadening understanding of Schur multipliers.

Findings

Exponent of Schur multiplier divides group exponent in these classes.

New proof techniques simplify existing results.

Extension of known properties to broader classes of p-groups.

Abstract

In this note, we give short proofs of the well-known results that the exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite $\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the same for a finite $\p$-group $\G$ satisfying $\G^{\p^2}\subset \gamma_{\p}(\G)$, and for $3$-groups of class $5$. We do this by proving a general lemma, and show that these three classes of groups satisfy the hypothesis of our lemma.