The Cayley's theorem says that every group $G$ is a subgroup of some symmetric group. More precisely, if $G$ is a group of order $n$, then $G$ is a subgroup of $S_n$.
In the course on group theory, this theorem is taught without applications. I came across one interesting application:
If $|G|=2n$ where $n$ is odd, then $G$ contains a normal subgroup of order $n$.
Q. What are the other elementary applications of Cayley's theorem in group theory, which can be explained to the undergraduates?
Best Answer
Cayley's theorem has an important status in group theory even in the absence of explicit applications: it's a sanity check on the definition of a group.
Before anyone had the idea of writing down the axioms for an abstract group, people studied what one might call concrete groups (I think the actual term was "systems of substitutions" or something like that, though), namely collections of bijections of some set $X$ closed under composition and inverses, or equivalently subgroups of $\text{Sym}(X)$. Cayley's theorem tells you that every abstract group is a concrete group, so the abstract axioms of group theory capture the concrete phenomenon, namely concrete groups, that they were invented to capture.
Similarly, before anyone had the idea of writing down the axioms for an abstract manifold, people studied what one might call concrete manifolds, namely submanifolds of $\mathbb{R}^n$. Theorems like the Whitney embedding theorem tell you that every abstract manifold is a concrete manifold, so again we find that the abstract axioms of manifold theory capture the concrete phenomenon that they were invented to capture.
See also, for example, this question where I asked what "Cayley's theorem for Lie algebras" should be.