# Finitary symmetric group is locally inner automorphism-balanced in symmetric group

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Finitary symmetric group (?)) satisfying a particular subgroup property (namely, Locally inner automorphism-balanced subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Statement

Suppose is a set, is the symmetric group on , and is the finitary symmetric group on , viewed as a subgroup of . Then, is a locally inner automorphism-balanced subgroup of . In other words, for any , the restriction of the inner automorphism of to is a locally inner automorphism of , i.e., for any finite subset of , there exists such that for all .