Let H be a subgroup of G and let X be the
set of left cosets of H in G. Show that there exists a normal subgroup N of G
such that $N \subseteq H$ and $G/N$ is isomorphic to a subgroup of $S_x$. (Suggestion:
Define a mapping $ \psi: G->S_x $ by $\psi(g) = f_g$, where $f_g(aH)= gaH$ for all $aH \in X$.
Show that $\psi$ is a homomorphism and that $ker(\psi) \subseteq H$
I have been able to show that the given $\psi$ is a homomorphism and that $\psi(G)$ is a subgroup of $S_x$ which makes $\psi: G -> \psi(G)$ an onto homomorphism which enables me to apply first isomorphism theorem and $ker(\psi)$ is the required N- normal subgroup. What I am unable to prove is $ker(\psi) \subseteq H$.
Best Answer
If $\psi(g)$ is in the kernel then it fixes every coset of $H$, in particular $H$ itself. But expanding your definitions, that means $$H=\psi(g)(H)=gH$$
Since $g=ge$ is in the RHS, it is in the LHS too, which means $g\in H$.