On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite $p$-Group

Details On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite $p$-Group On a Conjecture of a Bound for the Exponent of the Schur Multiplier of a Finite $p$-Group

2010-11-11

arxiv.org/abs/1011.2593v1

Mathematics

Group Theory

8 pages, to appear in Bulletin of the Iranian Mathematical Society

Scientific paper

Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $\exp(G)$ and let $m=\lfloor\log_pk\rfloor$. We show that $\exp(M^{(c)}(G))$ divides $\exp(G)p^{m(k-1)}$, for all $c\geq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $\exp(M(G))$ divides $\exp(G)$ for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement of some previous bounds for the exponent of $M^{(c)}(G)$ given by M. R. Jones, G. Ellis and P. Moravec in some cases.

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