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!

Consider the group $\langle \Bbb Z, +\rangle$ of integers under addition.
The Cayley representation of this group contains permutations $P_a$ where $$P_a(n) = n+a.$$  That is, each element of the Cayley representation is a permutation of $\Bbb Z$ obtained by shifting the entire number line rigidly left or right . (Or, in the case of $P_0$, leaving it fixed.)
Clearly, there are many permutations of $\Bbb Z$ that are not of this form.  For example, one such permutation exchanges $17$ and $23$ and leaves all the other integers fixed; this is not an element of the Cayley representation of 
$\langle \Bbb Z, +\rangle$.
