I am a student studying semi-direct products for the first time, and have this question:say $G = N \rtimes H$, where $N$ is normal and $H$ is another subgroup that "acts" on $N$. The quotient $G/N \cong H$. Would it be correct to say that if $K$ is a subgroup of $G$ containing $N$ then $K/N > 1$ if and only if $K \cap H > 1$. The way it seems to me is that if $K$ is a subgroup of $G$ containing $N$ then the size of $K/N$ is a measure of the intersection of $K$ with $H$, if there's any "excess" elements of $K$ outside $N$ then they must cntain some from $H$.
Is this correct?
Thanks, Kerim
Here is a very simple fact that holds for all subgroups $A$, $B$, and $C$. If $B \leq A$, then $A \cap BC = B(A \cap C)$.
So suppose that $G = NH$, where $N \cap H = 1$ and $N$ is a normal subgroup. Let $K$ be a subgroup such that $N \leq K$. Then by the above result
$$K = K \cap NH = N(K \cap H)$$
Now applying an isomorphism theorem shows $$K / N \cong K \cap H / (N \cap (K \cap H)) = K \cap H / (N \cap H)$$
Since $N \cap H = 1$, it follows that $K/N \cong K \cap H$.
Does this answer your question?