Inner automorphism

Symbol-free definition
An automorphism of a group is termed an inner automorphism if it can be expressed as defining ingredient::conjugation by an element of the group.

Note that the choice of conjugating element is not unique, in fact the possibilities for the conjugating element form a coset of the center.

Definition with symbols
An automorphism $$\sigma$$ of a group $$G$$ is termed an inner automorphism if there is an element $$g$$ in $$G$$ such that for all $$x \in G$$, $$\sigma(x) = c_g(x) := gxg^{-1}$$.

Note that the choice of $$g \in G$$ such that $$c_g = \sigma$$ need not be unique. In fact, the possibilities for $$g$$, for any $$\sigma$$, form a coset of the center of $$G$$.

Convention
If the convention we choose is of left actions, then the inner automorphism $$x \mapsto gxg^{-1}$$ is denoted as $$c_g$$, and is termed the inner automorphism induced by $$g$$ (or conjugation by $$g$$). It is also sometimes denoted as $${}^gx$$.

If the convention is to make the group act on the right, the inner automorphism induced by $$g$$ is defined as $$x \mapsto g^{-1}xg$$, and is denoted as $$x^g$$. Note that conjugation by $$g$$ in one convention equals conjugation by $$g^{-1}$$ in the other convention.

Justification for the definition
The notion of inner automorphism makes good sense because of the following fact: a group acts on itself as automorphisms via the conjugation map. This has the following consequences:


 * Every conjugation actually defines an automorphism
 * There is a homomorphism from the group to its automorphism group that sends each element to the corresponding conjugation map.

Homomorphism from the group to its automorphism group
The kernel of the natural homomorphism from a group to its automorphism group is the center of the group. This is because the condition that conjugation by an element be the identity map is equivalent to the condition that it commute with every element. The center of a group $$G$$ is denoted as $$Z(G)$$. The image, which is the inner automorphism group, is thus $$G/Z(G)$$.

Equivalence relation on elements
Two elements in a group are termed conjugate if they are in the same orbit under the action of the group by conjugation. The equivalence classes are termed conjugacy classes.

Equal to extensible automorphism
An automorphism of a group is inner if and only if it can be extended to an automorphism for any group containing that group. In other words, an automorphism is inner if and only if it is extensible to all groups. Analogous results hold when we restrict to groups satisfying certain properties.

An automorphism of a group is inner if and only if it can be pulled back to an automorphism for any surjective homomorphism to that group from another group. In other words, an automorphism is inner if and only if it is quotient-pullbackable to all groups. Analogous results hold when we restrict to groups satisfying certain properties.

Formalisms
Viewing the variety of groups as a variety of algebras, the inner automorphisms are precisely the I-automorphisms: the automorphisms expressible using a formula that is guaranteed to always yield an automorphism.

Metaproperties
A composite of inner automorphisms is inner, and an inverse of an inner automorphism is inner. The identity map is clearly inner. Hence, the inner automorphisms form a subgroup of the automorphism group, termed the inner automorphism group. This follows from the fact that group acts as automorphisms by conjugation.

In fact, they form a normal subgroup of the automorphism group.

If $$G \le H$$ are groups and $$\sigma$$ is an inner automorphism of $$G$$, then there exists an inner automorphism $$\sigma'$$ of $$H$$ such that the restriction of $$\sigma'$$ to $$G$$ is $$\sigma$$.

The idea is to take any conjugating candidate for $$\sigma$$ and consider the corresponding conjugation in the whole of $$H$$.

If $$\rho: G \to H$$ is a homomorphism of groups, and $$\sigma$$ is an inner automorphism of $$G$$, then there exists an inner automorphism $$\sigma'$$ of $$H$$ such that $$\rho \circ \sigma = \sigma' \circ \rho$$.

The idea is to take any conjugating candidate $$g$$ for $$\sigma$$ and define $$\sigma'$$ as conjugation by $$\rho(g)$$.

If $$\rho:G \to H$$ is a surjective homomorphism of groups, and $$\sigma$$ is an inner automorphism of $$H$$, there exists an inner automorphism $$\sigma'$$ of $$G$$ such that $$\rho \circ \sigma' = \sigma \circ \rho$$.

The idea is to take any conjugating candidate $$g$$ for $$\sigma$$, pick any inverse image of $$g$$ via $$\rho$$, and consider conjugation by that inverse element.

If $$G_1$$ and $$G_2$$ are two groups, and $$\sigma_1$$ and $$\sigma_2$$ are inner automorphisms on $$G_1$$ and on $$G_2$$ respectively, then $$\sigma_1 \times \sigma_2$$ is an inner automorphism on $$G_1 \times G_2$$. 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.

The idea is to take $$g_1, g_2$$ as conjugating candidates for $$\sigma_1, \sigma_2$$. Then the element $$(g_1,g_2)$$ serves as a conjugating candidate for $$\sigma_1 \times \sigma_2$$.

Textbook references

 * , Page 134 (formal definition, along with definition of the inner automorphism group)
 * , Page 14 (definition introduced in paragraph)
 * , Page 26 (Robinson uses the right action convention for inner automorphisms)
 * , Page 71 (formal definition)
 * , Page 26 (formal definition, after the notion of conjugation by an element)
 * , Page 175, Definition 3.2.9 (along with automorphism, formal definition)
 * , Page 90-91, (definition introduced in the context of Corollary 4.7(i))
 * , Page 123
 * , Page 68 (definition introduced in paragraph)
 * , Page 195, Exercise 2(c) of Miscellaneous Problems (definition introduced in exercise)