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!

Hint $ $ In $\rm\,\langle \Bbb Z/n, +\rangle\,$ if cycle $(0,1)$ is a shift $\rm\:x\to x\!+\!a\:$ then $\rm\: 0\!+\!a\equiv1,\,\ 1\!+\!a\equiv 0\:\Rightarrow\:1\equiv a\equiv -1.$
