# Finitary symmetric group is characteristic 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, Characteristic subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Statement

The Finitary symmetric group (?) on an infinite set is a Characteristic subgroup (?) of the Symmetric group (?) on that set.

## Facts used

- Finitary alternating group is monolith in symmetric group
- Monolith is characteristic
- Finitary symmetric group equals center of symmetric group modulo finitary alternating group
- Characteristicity is quotient-transitive

## Proof

The proof follows directly by combining facts (1)-(4).**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]