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

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 $G\to H.$ (Precise statement in $\S$1.) The group of such extended systems of automorphisms, unlike the group of inner automorphisms of $G$ itself, 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 the units of $K.$

If one looks at the above functorial extendibility property for endomorphisms, rather than just automorphisms, 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 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 open questions are noted.