# Subset containment gives inclusion of symmetric groups

From Groupprops

## Statement

Suppose is a subset of . Then, the symmetric group on can be identified with a subgroup of the symmetric group on as follows:

For any permutation of , we treat as a permutation of by setting it as the identity map on all elements in .

In particular, the symmetric group on the subset is the subset of the symmetric group on comprising those permutations that fix every element in .

## Examples

### An ordinary example

Suppose and . Suppose is the permutation of given by:

.

Then, we can treat as a permutation of simply by defining it as the identity map on the elements and :

.

### Extreme examples

- The empty subset is a subset of every set. The symmetric group on the empty set, which is the trivial group, is thus naturally a subgroup of the symmetric group on every set.
- Every set is a subset of itself. The corresponding subgroup of the symmetric group is the whole group.
- The symmetric group on a subset that is the complement of one element is precisely the stabilizer of that element. For instance, in the set , the symmetric group on the subset is precisely the stabilizer of in the whole symmetric group.