# Intersection of subgroups with normal series of prime index

Let $G$ be a group, and $H$ a subgroup such that for a prime $p$, there is a normal series

$$H = H^0 \triangleleft H^1 \triangleleft \cdots \triangleleft H^n = G$$

between $H$ and $G$ such that for all $i$, $[G_i : G_{i-1}] = p$. Call such a subgroup special.

Claim: Let $H$ and $K$ be two special subgroups. Then their intersection $H \cap K$ is special as well.

I've been working on this problem and have yet to make much progress. The case where $H$ and $K$ are normal subgroups is doable, since one can look at their quotients (which are $p$-groups), but I cannot see a way of generalizing from there. Indeed, finding a normal subgroup $N \subseteq H \cap K$ whose index in $G$ is a $p$-power would also work, but I don't see a way of constructing one. I also tried intersecting the normal series of $H$ with the normal series of $K$ to construct one for their intersection, but it quickly got very complicated, and I don't see a way forward.

Any tips would be appreciated.

• Have you tried building first a series from $H\cap K$ to $H$, and then from $H$ to $G$? – Steve D Nov 21 '17 at 3:19
• @SteveD Nope, but that's a really great idea! Thanks! – Mr Bingley Nov 21 '17 at 4:03