Self-normalizing implies canonical Abelianization

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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