Inner-centralizing automorphism

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.