Inner-centralizing automorphism

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This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Definition

An automorphism \sigma of a group G is termed an inner-centralizing automorphism if it satisfies the following equivalent conditions:

  1. It is in the centralizer (inside the automorphism group of G) of the inner automorphism group of G, i.e., it commutes with every inner automorphism of G.
  2. It preserves every coset of the center of G, and hence induces the identity map on the quotient group of G by its center. In other words, it is in the kernel of the natural map \operatorname{Aut}(G) \to \operatorname{Aut}(G/Z(G)).

The inner-centralizing automorphisms of a group arise from homomorphisms from the abelianization of G to the center of G. If \alpha is such a homomorphism, we can attempt to define a homomorphism as follows:

g \mapsto \alpha(\overline{g}) g

where \overline{g} is the coset of the derived subgroup of G containing g. Such a map is always an endomorphism; however, it need not be an automorphism. If there is an element in the kernel, it must be in the center of G. If we denote by \theta the map induced on the center by g \mapsto \alpha(\overline{g}), we obtain that the map is an automorphism iff g \mapsto g\theta(g) is an automorphism of the center.