Thread: normal p-subgroups of a finite group

1. normal p-subgroups of a finite group

Let $G$ be a finite group. Let $H$ and $K$ be normal $p$-subgroups of $G$ for some prime $p$ with $H$ being a proper subgroup of $K$ and $H/K$ is elementary abelian p-group. Let $S$ be a normal subgroup of $K$ with $H. If $H$ contain every element of order $p$ of $S$ and $K=$, then $H$ contain every element of order $p$ of $K$.

I need to prove the above statement. Thanks in advance.

2. Re: normal p-subgroups of a finite group

your question doesn't make sense, as written. if H < K, then what does H/K mean?

3. Re: normal p-subgroups of a finite group

I fix the mistake in the question with some additions.
Let $G$ be a finite solvable group. Let $K/H$ be a chief factor of $G$ that is not of prime order, where $K$ is a $p$-subgroup of $G$ for some prime $p$ divides the order of $G$. Let $S$ be a proper normal subgroup of $K$ with $H and $|S/H|=p$. If $H$ contain every element of order $p$ of $S$ and $K=$, then $H$contains every element of order $p$ of $K$.

I need to prove the above statement.

Here is what I know.

Since $G$ is solvable then $K/H$ is abelian $p$-group of exponent $p$. $\bigcap_{g \in G}S^{g}$ is a normal subgroup of $G$that contains $H$. So H= $\bigcap_{g \in G}S^{g}$.