Conjugacy and Stabilizers in Group Actions

[fishturtle1]

Homework Statement

The problem is two parts:

i) Let $X$ be a finite $G$-set , and let $H \le G$ act transitively on $X$. Then $G = HG_x$ for each $x \in X$.

ii) Show that the Frattini argument follows from i).

Relevant Equations

Relevant equations:

Frattini Argument: Let $K$ be a normal subgroup of a finite group $G$. If $P$ is a Sylow $p$-subgroup of $K$ (for some prime $p$), then $$G = KN_G(P).$$

$X$ is a finite $G$-set means that $G$ acts on $X$ and both $X$ and $G$ are finite.

$G_x = \lbrace g \in G : g\cdot x = x \rbrace$

Attempt at solution:

Proof of i): Let $x \in X$. Its clear $G \supseteq HG_x$. Let $g \in G$.Then there is $y \in X$ such that $g \cdot x = y$. Since $H$ acts transitively on $X$, there is $h \in H$ such that $h \cdot x = y$. So, $g \cdot x = h \cdot x$. This gives $$(h^{-1}g)\cdot x = x$$ Hence, $g = h(h^{-1}g) \in HG_x$ and we can conclude $G = HG_x$. []

For ii), let $X = \lbrace gPg^{-1} : g \in G\rbrace$. Then $G$ acts on $X$ by conjugation and $N_G(P)$ is the stabilizer of $P$. But I'm not sure if $K$ acts transitively on $X$. I know I haven't used the fact that $K$ is a normal subgroup of $G$. Can I have a hint on how to solve ii), please?

 

[fishturtle1]

I think I got it?

Proof of ii): Let $X = \lbrace gPg^{-1} : g \in G \rbrace$. Then $G$ acts on $X$ by conjugation. If $g \in G$, then $gPg^{-1} \le gKg^{-1} = K$ since $K$ is a normal subgroup of $G$. Hence, $gPg^{-1}$ is a Sylow p-subgroup of $K$. So there is $k \in K$ such that $kPk^{-1} = gPg^{-1}$. In other words, $K$ acts transitively on $X$. Moreover, $N_G(P)$ is the stabilizer of $P$. By i),
$$G = KN_G(P).$$
[]