The set of all permutations on a set is denoted by .
Under the binary operation of function composition, is a group
and is called the *symmetric group* on .
In the case that then we customarily write the
symmetric group as and talk about
the *symmetric group of degree n*.
The order of is

The set of even permutations in forms a subgroup.
This subgroup is called the *Alternating Group* and is denoted by
. Its order is *n*!/2.

A *permutation group* is a subgroup of a symmetric group on some
set .

Recall that the cycle (1 2 3) can also be written as (3 1 2) or
(2 3 1). In general, there are *n* ways to write a cycle of length
*n*.
We can list the types of permutations in and count the number
of each type. By the shape of a cycle we mean its length and not the
particular integers that are used. We will refer to such a cycle as
(. . .), say, for a cycle of length 3. The shape of a permutation
will be the different combinations of cycle shapes that can be used.
For example, in , a permutation written as a
product of disjoint cycles must have one of the following shapes:

**shape 1.**- (. . . . .) where '.' represents a position to be filled with an integer between 1 and 5 inclusive.
**shape 2.**- (. . . .)
**shape 3.**- (. . .)
**shape 4.**- (. .)
**shape 5.**- (. . .)(. .)
**shape 6.**- (. .)(. .)
**shape 7.**- IDENTITY ELEMENT

In total then, we have accounted for 1+24+30+20+10+20+15 which is 120 or 5!. Note then that has elements of order 1, 2, 3, 4, 5 and 6. Also, of these shapes, only shapes 1, 3, 6 and 7 are elements of .

Sun Mar 30 14:48:35 PST 1997