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**Chandru1** Let $\displaystyle G$ be a finite group in which $\displaystyle (ab)^{p}=a^{p}b^{p}$ for some prime $\displaystyle p$ dividing the order of $\displaystyle G$. Prove that if $\displaystyle P$ is the p-sylow subgroup of $\displaystyle G$, then there exists a normal subgroup $\displaystyle N$of $\displaystyle G$ with $\displaystyle P \cap N=(e)$ and $\displaystyle PN=G$.

All i can see is that the mapping $\displaystyle x \mapsto x^{p}$ is a homomorphism and the set $\displaystyle S=\{ x^{p} \ | \ x \in G\}$ and $\displaystyle K=\{ x^{p-1} \ | \ x \in G\}$ are normal and have trivial intersection.