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**doug** Let G be a finite group, $\displaystyle \pi$ a set of primes, $\displaystyle \Omega$ the set of normal $\displaystyle \pi$-subgroups of G, and $\displaystyle \Gamma$ the set of normal subgroups X of G with G/X a $\displaystyle \pi$-group. Prove

(1) If H, K $\displaystyle \in \Omega$ then HK $\displaystyle \in \Omega$. Hence $\displaystyle \langle \Omega \rangle$ is the unique maximal member of $\displaystyle \Omega$.

(2) If H, K $\displaystyle \in \Gamma$ then $\displaystyle H \cap K \in \Gamma$. Hence $\displaystyle \bigcap_{H\in\Gamma} H$ is the unique minimal member of $\displaystyle \Gamma$

I would really appreciate your help.