# Subgroup generated by commutator of generators of free group on two generators is automorph-conjugate

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Automorph-conjugate subgroup (?)) in a particular group or type of group .

## Statement

Let be a free group on two generators, with being the generators. Let be the subgroup of generated by the commutator :

.

Then, is an automorph-conjugate subgroup of .

## Facts used

- Automorph-conjugate iff conjugate to image under a generating set of automorphism group
- Elementary Nielsen automorphisms generate the automorphism group of a finitely generated free group

## Proof

**Given**: is a free group with freely generating set . .

**To prove**: is automorph-conjugate in .

**Proof**: By fact (2), the elementary Nielsen automorphisms of generate . We use a modified version of this generating set to show that is automorph-conjugate in via fact (1):

- Replacing by its inverse: .
- Replacing by its inverse: .
- Swapping and : .
- Replacing by : .