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.