An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

TL;DR

This paper characterizes inner automorphisms and endomorphisms across algebraic structures using functorial properties, revealing that inner automorphisms are uniquely determined, while inner endomorphisms can exhibit more complex behaviors.

Contribution

It introduces a functorial framework for understanding inner automorphisms and endomorphisms in groups and algebras, highlighting differences and new phenomena.

Findings

Inner automorphisms are characterized functorially within the category of groups.

The group of functorial automorphisms is isomorphic to the original group.

Inner endomorphisms can be more complex, especially in algebraic contexts.

Abstract

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo units of K. If one substitutes "endomorphism" for "automorphism" in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism" in the literature, some overlapping the one introduced here, are noted; the concept of an inner {\em derivation} of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a "co-inner" endomorphism is briefly examined. Several questions are posed.