Class-preserving automorphism

Definition
An defining ingredient::automorphism of a group is termed a class-preserving automorphism or class automorphism if it takes each element to within its defining ingredient::conjugacy class. In symbols, an automorphism $$\sigma$$ of a group $$G$$ is termed a class automorphism or class-preserving automorphism if for every $$g$$ in $$G$$, there exists an element $$h$$ such that $$\sigma(g) = hgh^{-1}$$. The choice of $$h$$ may depend on $$g$$.

Related properties

 * Subgroup-conjugating automorphism:
 * Class-inverting automorphism

Facts

 * Class-preserving automorphism group of finite p-group is p-group

Metaproperties
Clearly, a product of class automorphisms is a class automorphism, and the inverse of a class automorphism is a class automorphism. Thus, the class automorphisms form a group which sits as a subgroup of the automorphism group. Moreover, this subgroup contains the group of inner automorphisms, and is a normal subgroup inside the automorphism group.

Let $$G_1$$ and $$G_2$$ be groups and $$\sigma_1, \sigma_2$$ be class automorphisms of $$G_1, G_2$$ respectively. Then, $$\sigma_1 \times \sigma_2$$ is a class automorphism of $$G_1 \times G_2$$.

Here, $$\sigma_1 \times \sigma_2$$ is the automorphism of $$G_1 \times G_2$$ that acts as $$\sigma_1$$ on the first coordinate and $$\sigma_2$$ on the second.