# Finitary symmetric group is not fully invariant 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 (?))notsatisfying a particular subgroup property (namely, Fully characteristic subgroup (?)) in a particular group or type of group (namely, Symmetric group (?)).

## Contents

## Statement

The Finitary symmetric group (?) on an infinite set is *not* fully characteristic as a subgroup of the symmetric group.

## Related facts

- Finitary symmetric group is characteristic in symmetric group
- Finitary symmetric group is strictly characteristic in symmetric group
- Finitary symmetric group is not I-characteristic in symmetric group

## Proof

### Proof idea

We use the fact that an infinite set can be put in bijection with a union of countably many copies of that set. Using this bijection, we construct an endomorphism that sends each permutation to a permutation that acts like it on *each* of the countably many copies. This is an injective endomorphism, and under this endomorphism, every non-identity permutation gets mapped to a permutation that is not finitary. In particular, the finitary symmetric group is sent to a subgroup that it intersects trivially.