= Solution
Choose $x_0\in X$ with <stabilizer subgroup> $H$. Transitivity means that every point is $gx_0$ for some $g$. Its <stabilizer subgroup> is
$$
G_{gx_0}=gHg^{-1},
$$
since $a(gx_0)=gx_0$ exactly when $g^{-1}ag\in H$. The <kernel of a group homomorphism> arising from the <group action> is the set of elements fixing every point, hence
$$
\boxed{\ker(G\to\operatorname{Sym}(X))=\bigcap_{g\in G}gHg^{-1}=\operatorname{Core}_G(H).}
$$
This is the <subgroup core>. A <group homomorphism> is injective precisely when its kernel is trivial. The desired equivalence is therefore exactly the equivalence between a trivial <subgroup core> and a <faithful group action>. No finiteness assumption is required in this part.
Back to article page
