Subset containment gives inclusion of symmetric groups
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 .
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 :
- 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.