2Symmetric group I

IA Groups

2 Symmetric group I
We will devote two full chapters to the study of symmetric groups, because
it is really important. Recall that we defined a symmetry to be an operation
that leaves some important property of the object intact. We can treat each
such operation as a bijection. For example, a symmetry of
R
2
is a bijection
f
:
R
2
R
2
that preserves distances. Note that we must require it to be a
bijection, instead of a mere function, since we require each symmetry to be an
inverse.
We can consider the case where we don’t care about anything at all. So a
“symmetry” would be any arbitrary bijection
X X
, and the set of all bijections
will form a group, known as the symmetric group. Of course, we will no longer
think of these as “symmetries” anymore, but just bijections.
In some sense, the symmetric group is the most general case of a symmetry
group. In fact, we will later (in Chapter 5) show that every group can be written
as a subgroup of some symmetric group.