On a conjecture of a bound for the exponent of the Schur multiplier of a finite $p$-group

Abstract

Let $G$ be a $p$-group of nilpotency class <br />$k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides <br />$exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that <br />$exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result <br />is an improvement of some previous bounds for the exponent of <br />$M^{(c)}(G)$ given by M. R. Jones, G. Ellis and P. Moravec in some cases.