We now give one of the most important results in group theory. It shows that
every group is isomorphic to a group of permutations. That is, every group
is isomorphic to a subgroup of the symmetric group on some suitable set 
.
Cayley's Theorem Every group is isomorphic to a group of permutations.
PROOF: The proof of this result is constructive in that it gives you an
algorithm for converting a group into a permutation group.
Let G be a group and let 
be the symmetry group of G
(thought of as a set). For each 
let 
be the permutation
defined by
In other words, is derived from right multiplication of the
elements of G by g. That
is a permutation is easily established.
For (i), 
shows it is onto and (ii) 
if and only if xg=yg if and only if x=y shows that it is one-to-one.
Let 
and define 
by 
.
Then 
. Now for any 
we have
Hence, 
and 
,
proving f is a homomorphism. Clearly f is onto H. If 
, then
for all we have 
. Therefore, 
and f is
one-to-one. Thus, H is a subgroup of and isomorphic to G.
Consequently, if G is finite of order n then G is isomorphic to a
subgroup of 
. It is possible, as we shall see later on, that G is
isomorphic to a subgroup of 
where k<n. This raises the question of
what is the smallest positive integer r so that G is isomorphic to
a subgroup 
? As an example, consider the symmetry group of the
equilateral triangle, 
. Cayley's Theorem would yield a representation of the
elements as permutations of 
. However, if we label the
vertices of the triangle with the numbers 1, 2 and 3, then we see that the
elements may be represented as permutations of 
. With one certain
labelling, we would get the rotation through 
to correspond to
and the rotation about the y-axis to be 
.