Subset containment gives inclusion of symmetric groups

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Statement

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

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

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

Examples

An ordinary example

Suppose A = \{ 1,2,3 \} and B = \{ 1,2,3,4,5 \}. Suppose \sigma is the permutation of A given by:

\sigma(1) = 3, \sigma(2) = 1, \sigma(3) = 2.

Then, we can treat \sigma as a permutation of B simply by defining it as the identity map on the elements 4 and 5:

\sigma(1) = 3, \sigma(2) = 1, \sigma(3) = 2, \sigma(4) = 4, \sigma(5) = 5.

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 \{ 1,2,3,4 \}, the symmetric group on the subset \{ 1,3,4 \} is precisely the stabilizer of 2 in the whole symmetric group.