In "A Book of Abstract Algebra" by Pinter, chapter 16.G is the following exercise:
If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element $a \in G$, define $ρ_a:X \to X$ as follows: $$ρ_a(xH)=(ax)H$$ 1 Prove that each $ρ_a$ is a permutation of X
2 Prove that $h:G \to S_X$ defined by $h(a)=ρ_a$ is a homomorphism
3 Prove that the set $\{a \in H:xax^{-1} \in H$ for every $x \in G\}$, that is, the set of all elements of H whose conjugates are all in H, is the kernel of h
4 Prove that if H contains no normal subgroup of G except {e} (trivial group), then G is isomorphic to a subgroup of $S_X$
I finished proving these, but I don't understand what the theorem is really saying, and why it's an interesting result and what "sharper" means, since the original Cayley's theorem already said that every group G is isomorphic to a group of permutations, and here I proved that G is isomorphic to "another" group of permutations, but now making extra assumptions.
The wikipedia page on Cayley's theorem has the same result but there instead of making the assuption that H has no normal subgroups except for the trivial group, it just says that the case where H itself is the trivial group produces the original Cayley's theorem, but I still can't "see" it, is the fact that the quotient group of G by the set in point 3 (or the normal core of H in G) is isomorphic to a group of permutations important?.
I understand that Cayley's theorem is one "case" of this new theorem but I'd like to deeply understand what this new one means and how to intuitively think about it.
It is sharper because the degree (i.e. cardinality of the set upon the group acts) is smaller if $H\ne 1$. Cayley's theorem embeds $G$ into $S_n$ where $n=|G|$ while this exercise embeds $G$ into $S_n$ with $n=|G:H|$. For instance, Cayley says that $S_4$ is a subgroup of $S_{24}$. But taking $H=S_3$, the sharper version states that $S_4$ is a subgroup of $S_4$.