How is the "Representation of Cosets theorem" a generalization of Cayley's theorem? 
Theorem 2.88 (Representation on Cosets). Let $G$ be a group, and let $H$ be a subgroup of $G$ having finite index $n$. Then there exists a homomorphism $\varphi: G \rightarrow S_n $ with $\ker \varphi \leq H$.

The author claims this is a more interesting generalization of Cayley's theorem. However this doesn't seem to really tell anything? If $\ker \varphi \leq H$ it could be the case that ker $\varphi = \{1\}$, which is Cayley's theorem, so isn't this statement weaker than Cayley's theorem?
 A: What you've observed is that Cayley's theorem is a special case of this theorem: if $G$ is finite then we may pick $H =\{1\}$, which forces $\ker \varphi = \{1\}$, giving the conclusion of Cayley's theorem. But of course the theorem is more general than this; $G$ need not be finite and $\ker \varphi$ need not be trivial. That's what Rotman means by "more interesting generalization".
Saying that this is weaker than Cayley's theorem would mean that Cayley's theorem implies this result, which is not the case (of course there's no way to make this precise because both theorems are true).
A: The only reason to restrict this to subgroups of finite index is that the definition of the symmetric group for an infinite set is not a given. For some, the symmetric group on an infinite set, $S_X$, just means all bijections $X\to X$; others require the bijections to have finite support (that is, $\mathrm{supp}(\sigma) = \{x\in X\mid\sigma(x)\neq x\}$ is finite for each bijection $\sigma$), so that the elements of $S_X$ can still be described as consisting of a product of disjoint cycles, etc. (Also, its most common application is that a subgroup of finite index contains a normal subgroup of finite index, so the more general statement does not provide wider applications).
If you simply define $S_X$ to be the group of bijections $X\to X$, a group under composition, then this theorem holds:

Theorem. Let $G$ be a group and let $H$ be a subgroup of $G$. Then there exists a homomorphism $\varphi\colon G\to S_{G/H}$ (where $G/H$ is the set of left cosets of $H$ in $G$) with $\ker(\varphi)\leq H$.

The special case of this theorem which yields Cayley’s Theorem is $H=\{e\}$.
A: In general, the kernel of a $G$-action on a set $X$ (i.e. the kernel of the equivalent homomorphism $\lambda:G\to\operatorname{Sym}(X)$) is given by $\operatorname{ker}\lambda=\bigcap_{x\in X}\operatorname{Stab}(x)$.
In particular, if we take the $G$-action by left multiplication on the set $X:=\{gH,g\in G\}$, then there is a homomorphism $\lambda:G\to \operatorname{Sym}(G/H)\cong S_{[G:H]}$, with  $\operatorname{ker}\lambda=\bigcap_{g\in G}gHg^{-1}\le H$ (see e.g. here).
This result, along with the First Homomorphism Theorem, shows that, among all the subgroups $H$ of a given index in $G$, an eventual normal one provides the "best focus" (literally, i.e. the smallest image) of $G$ in $S_{[G:H]}$. In fact, in this case: $\operatorname{ker}\lambda=H=(\operatorname{ker}\lambda)_{\operatorname{max}}$.
To recap:
Let's define $\mathcal{H}_k:=\{H\le G\mid [G:H]=k\}$. Then, $\forall H\in\mathcal{H}_k$, the group $Q_G(H):=G/\bigcap_{g\in G}gHg^{-1}$ embeds into $S_k$. Moreover, if $\exists \tilde H\in \mathcal{H}_k$ such that $\tilde H\unlhd G$, then $|\operatorname{im}(Q_G(\tilde H))|=|\operatorname{im}(G/\tilde H)|=\operatorname{min}\{|\operatorname{im}(Q_G(H))|, H \in \mathcal{H}_k\}$.
