$a)$Show that there is a nontrivial group homomorphism $ φ : G →S_{k}.$ and show that G has a nontrivial proper normal subgroup

Question

Let G be a group of order n, H a subgroup of G of order m, $k =\frac{n}{m}$ and $S_{k}$ the symmetric group on $ k$ symbols

$a)$Show that there is a nontrivial group homomorphism $ φ : G →S_{k}.$

$b)$ Assuming $ \frac{k!}{2}< n$, show that G has a nontrivial proper normal subgroup

My idea : i was thinking about fisrt theorem of isomorphism

Pliz help me ,,as im very much confused....

Answer 1

Consider the equivalence relation defined on $G$ by $x\simeq y$ iff $x=yh, h\in H$, the cardinal of the set $G/H$ of equivalence classes is $k$ and $G$ acts on $G/H$ by $g(xH)=(gx)H$ by an action defined by $f:G\rightarrow S_k$ This action is not trivial sincd $G$ acts transitively.

Suppose that $k!/2<n$, if $f$ is not surjective, the image of $f$ is a subgroup $L$ and $card(L)$ divides $k!$ this implies that $card(L)\leq k!/2$ and the kernel is not trivial. If $f$ is surjective $f^{-1}(A_k)$ is a non trivial subgroup of $G$.