Revistes Catalanes amb Accés Obert (RACO)

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

George M. Bergman


The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those au-tomorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G ! H: (Precise statement in x1. ) 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 eld 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 briey examined. Several open questions are noted.

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