If $G$ is a group with a subgroup $H$ of finite index $n$, then $G$ has a normal subgroup $N$ whose index in $G$ is finite.
I found a proof of the question here: How to prove that if $G$ is a group with a subgroup $H$ of index $n$, then $G$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
I have a few questions regarding this proof:
- The map $\phi: G \to \text{Sym}(X)$ given by $\phi(x)(aH)=(xa)H$ takes an argument from $G$. Why do we have $(aH)$ behind $\phi(x)$? What does the notation mean? Can we write $\phi(x)=xG/H$ instead?
2.How do I show that $\phi: G \to \text{Sym}(X)$ given by $\phi(x)(aH)=(xa)H$ is a homomorphism? I don't see immediately that $\phi(xy)=\phi(x)\phi(y)$.