Cayley's Theorem question: examples of groups which aren't symmetric groups. Basically, Cayley's Theorem says that every finite group, say $G$, is isomorphic to a subgroup of the group $S_G$ of all permutations of $G$.
My question: why is there the word "subgroup of"? If we omit this word, is the statement wrong? brief examples would be nice.
Thank you guys so much!

The symmetric group $S_n$ has order $n!$ whereas there exists a group of any order (eg. $\mathbb{Z}_n$ has order $n$).
