Symmetric group on finite or cofinite subset is conjugacy-closed

This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup
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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Conjugacy-closed subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

Suppose $S\subseteq T$ are sets. Then, the symmetric group $\mathrm{S}\mathrm{y}\mathrm{m}(S)$ embeds naturally as a subgroup of $\mathrm{S}\mathrm{y}\mathrm{m}(T)$: any permutation of $S$ extends to a permutation of $T$ as the identity map on $T\setminus S$.

With this embedding, if either $S$ or $T\setminus S$ is finite, $\mathrm{S}\mathrm{y}\mathrm{m}(S)$ is a conjugacy-closed subgroup in $\mathrm{S}\mathrm{y}\mathrm{m}(T)$. In other words, if two elements of $\mathrm{S}\mathrm{y}\mathrm{m}(S)$ are conjugate in $\mathrm{S}\mathrm{y}\mathrm{m}(T)$, they are also conjugate in $\mathrm{S}\mathrm{y}\mathrm{m}(S)$.

Related facts

Stronger formulation

Symmetric group on finite or cofinite subset is subset-conjugacy-closed: Not only can we perform conjugation of single elements, we can also perform conjugation of subsets inducing exactly the same map on each element of the subset.

Breakdown for infinite coinfinite subsets

If both $S$ and $T\setminus S$ are infinite, then $\mathrm{S}\mathrm{y}\mathrm{m}(S)$ is not conjugacy-closed in $\mathrm{S}\mathrm{y}\mathrm{m}(T)$. Further information: Symmetric group on infinite coinfinite subset is not conjugacy-closed

Other related facts

Facts about conjugacy and conjugacy-closedness:

• Finitary symmetric group is conjugacy-closed in symmetric group
• Finitary symmetric group on subset is conjugacy-closed
• Brauer's permutation lemma: This states that the symmetric group is conjugacy-closed in the general linear group.
• Symmetric groups on finite sets are complete, Symmetric groups on infinite sets are complete: Any automorphism of a symmetric group is inner, except when the underlying set has size six. Further, the symmetric groups are centerless.

Some other facts about finitary symmetric groups and symmetric groups related to conjugacy:

• Finitary symmetric group on infinite subset is conjugate-dense
• Symmetric group on proper subset is not conjugate-dense