On the exponent conjecture of Schur

TL;DR

This paper proves the longstanding conjecture that the exponent of the second homology group divides the group exponent for specific classes of finite groups, using induction and improving existing bounds.

Contribution

It establishes the conjecture for various classes of finite groups and refines previous bounds through inductive methods.

Findings

Proved the conjecture for p-groups of class at most p

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

Improved bounds on group exponents and homology exponents

Abstract

It is a longstanding conjecture 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 $p$-groups of class at most $p$, 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. We achieve most of our results using an induction argument.