# Inner-centralizing automorphism

From Groupprops

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 of a group is termed an **inner-centralizing automorphism** if it satisfies the following equivalent conditions:

- It is in the centralizer (inside the automorphism group of ) of the inner automorphism group of , i.e., it commutes with every inner automorphism of .
- It preserves every coset of the center of , and hence induces the identity map on the quotient group of by its center. In other words, it is in the kernel of the natural map .

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

where is the coset of the derived subgroup of containing . 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 . If we denote by the map induced on the center by , we obtain that the map is an automorphism iff is an automorphism of the center.