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Let $G$ be a finite group and $M(G)$ be the Schur multiplier of $G$. Show that the exponent $\exp(G)$ of $G$ divides the product of the exponents $\exp(M(S_p))$ of the Sylow $p$-subgroups $S_p$ of $G$.

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    If you know that every Sylow $p$-subgroup of $M(G)$ is isomorphic with a subgroup of $M(S_p)$, then then $M(G)$ is isomorphic to the direct product of all of its Sylow $p$-subgroups, and you should be able to solve the problem.2012-10-27

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