On the Exponent Conjectures

Abstract

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$ or $6$. Furthermore we prove that if $G$ is a solvable group of derived length $d$ and $\e(G)=p$, then $\e(H_2(G,\mathbb{Z})) \mid (\e(G))^{d-1}$. We also show that if $G$ is a finite $2$ or $3$ generator group of exponent 5, then $\e(H_2(G,\mathbb{Z})) \mid (\e(G))^2$.