Tour:Subset containment gives inclusion of symmetric groups

This article adapts material from the main article: 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.

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