On the Exponent of the Schur Multiplier

TL;DR

This paper proves Schur's conjecture for specific classes of finite groups, including certain nilpotent and metabelian groups, and improves existing bounds on the exponent of the Schur multiplier.

Contribution

It establishes the conjecture for finite nilpotent groups of odd exponent and nilpotency class 5, p-central metabelian p groups, and groups studied by Wilson, while refining previous bounds.

Findings

Proves the conjecture for finite nilpotent groups of odd exponent and class 5

Validates the conjecture for p-central metabelian p groups

Improves bounds on the exponent of the Schur multiplier

Abstract

A longstanding problem attributed to I. Schur says that for a finite group $G$, the exponent of the second homology group $H_2(G, \mathbb{Z})$ divides the exponent of $G$. In this paper, we prove this conjecture for finite nilpotent groups of odd exponent and of nilpotency class 5, $p$-central metabelian $p$ groups, and groups considered by L. E . Wilson in \cite{LEW}. Moreover, we improve several bounds given by various authors.