$H$, $N$ are subnormal subgroups in the finite group $G$ and $G = H*N$. Show: $(H*N)^{\infty} = H^{\infty}*N^{\infty}$. (And $G^{\infty} := \bigcap\limits_{i\geq 0}G^{i}$, and $G^{i+1} = [G, G^{i}]$ from the lower central series.)
We got the hint to start with $H,N$ normal in $G$; prove this with induction and later tranfer this to the case where $H$ and $N$ are subnormal.
So I started: The case i=1 holds apperently.
For the induction step I'd need some help. I have so far:
Since $G$ is finite, we know that the central series stagnates at some point, let's say:
$\bigcap\limits_{i\geq 0}G^{i} = G^{m}$ for some $m$.
The same holds then for $H^{\infty}$ and $N^{\infty}$. I named them:
$H^{\infty} = H^{k}$ for some $k$ and $N^{\infty} = N^{l}$ for some $l$.
So I get $G^{\infty} = G^{m} = (H*N)^{m} \overset{?}{=} H^{\infty}*N^{\infty} = H^k*N^l$.
So I have to show that $m=k=l$ right?
Is this so far OK and a good way to approach the problem? I got at this point already stuck.. So I'd be very happy if someone had a little hint how to go on :)
All the best, Sara!