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Let $G$ be a $p$-group. Let $H$ be any subgroup of $G$. How to prove that there exists subgroups of $G$ such that $$H = H_0 \lt H_1 \lt H_2\lt \cdots \lt H_n=G$$ such that $|H_{i+1}/H_i|=p$?

I have proved the theorem when H is normal subgroup of G, & when H is trivial subgroup.

Here H is any given fixed subgroup to start with (not necessarily normal)

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    Can you clarify your question? By the way, for a $p$-group we know the center of $G$ is non-trivial, so that gives you (if the group is non-abelian) a normal subgroup, perhaps that is what you want ; but at first glance I cannot guess what you're actually looking for. And I removed the tag "galois-theory" because this is completely unrelated.2012-04-13

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