Self-normalizing implies canonical Abelianization

Statement

Property-theoretic statement

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

Verbal statement

Suppose $B$ is a self-normalizing subgroup of a group $G$. Then, the Abelianization of $B$ is determined upto canonical isomorphism in $G$. 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 $B$
• 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 $p$ to $p$ is the identity, and given $p,q,r$, the isomorphism from $p$ to $r$ is the composite of the isomorphisms from $p$ to $q$ and $q$ to $r$