Permutation groups

#matematik-groups

Definition: Permutation group

Let be a set. We then define the permutation group on as together with composition .

We can quite easily see that for any , is a Group under composition . However, it is not necessarily commutative since the compositions of bijections are not necessarily commutative; swapping the first two elements and then the second two among three elements is different from swapping the second two and then the first two.

Permutation groups are groups

Let be a set. Then is a group.

Proof:
If we know since the composition of two bijections is a bijection. Compositions are associative, and by definition invertible with the identity on as the unit element.

Notation

If , we define and call it the symmetric group of degree or the group of permutations of elements.