# Tour:Subset containment gives inclusion of symmetric groups

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**This article adapts material from the main article:** subset containment gives inclusion of symmetric groups

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## 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.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: cycle decomposition for permutations|UP: Introduction five (beginners)|NEXT: group action

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part