# Self-normalizing implies canonical Abelianization

From Groupprops

## Statement

### Property-theoretic statement

Any self-normalizing subgroup of a group possesses a canonical Abelianization.

### Verbal statement

Suppose is a self-normalizing subgroup of a group . Then, the Abelianization of is determined upto canonical isomorphism in . In other words, we can do the following, in a manner that is invariant under inner automorphisms:

- Consider the space whose points are the Abelianizations of conjugates of
- For any two points in the space, define a unique isomorphism between those points, such that the isomorphisms form a category i.e. the isomorphism from to is the identity, and given , the isomorphism from to is the composite of the isomorphisms from to and to