finite group (prove)

**doug** · September 1st 2011, 12:00 PM

Let G be a finite group, $\pi$ a set of primes, $\Omega$ the set of normal $\pi$ -subgroups of G, and $\Gamma$ the set of normal subgroups X of G with G/X a $\pi$ -group. Prove
(1) If H, K $\in \Omega$ then HK $\in \Omega$ . Hence $\langle \Omega \rangle$ is the unique maximal member of $\Omega$ .
(2) If H, K $\in \Gamma$ then $H \cap K \in \Gamma$ . Hence $\bigcap_{H\in\Gamma} H$ is the unique minimal member of $\Gamma$

I would really appreciate your help.

**doug** · September 2nd 2011, 01:59 PM

In other words we have to prove that $O_\pi (G)$ and $O^\pi (G)$ are well-defined, where $O^\pi (G)$ denotes the largest normal pi-subgroup of G and $O_\pi (G)$ denotes the smallest normal subgroup H of G such that G/H is a pi-group.
Thank you very much for your help in advance!

**NonCommAlg** · September 2nd 2011, 07:16 PM

Originally Posted by doug

Let G be a finite group, $\pi$ a set of primes, $\Omega$ the set of normal $\pi$ -subgroups of G, and $\Gamma$ the set of normal subgroups X of G with G/X a $\pi$ -group. Prove
(1) If H, K $\in \Omega$ then HK $\in \Omega$ . Hence $\langle \Omega \rangle$ is the unique maximal member of $\Omega$ .
(2) If H, K $\in \Gamma$ then $H \cap K \in \Gamma$ . Hence $\bigcap_{H\in\Gamma} H$ is the unique minimal member of $\Gamma$

I would really appreciate your help.

1) clearly $HK$ is a normal subgroup. now, since $|HK|=\frac{|H| \cdot |K|}{|H \cap K|},$ every prime divisor of $|HK|$ is either a prime divisor of $|H|$ or $|K|$ . thus, since $H$ and $K$ are $\pi$ -groups, $HK$ is a $\pi$ -group too.

2) define a group homomorphism $\varphi: G \longrightarrow G/H \times G/K$ by $\varphi(g)=(gH, gK).$ then $\ker \varphi =H \cap K$ and so $G/(H \cap K)$ is isomorphic to a subgroup of $G/H \times G/K.$ thus $|G/(H \cap K)|$ divides $|G/H| \cdot |G/K|$ and you're done.

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