Inner 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)
View other automorphism properties OR View other function properties

Definition

Symbol-free definition

An automorphism of a group is termed an inner automorphism if it can be expressed as 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 $σ$ of a group $G$ is termed an inner automorphism if there is an element $g$ in $G$ such that for all $x \in G$ , $σ(x) = g x g - 1$ .

Note that the choice of $g \in G$ such that $c g = σ$ need not be unique. In fact, the possibilities for $g$ , for any $σ$ , 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 $g$ ).

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 $x g$ .

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.

Facts

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.

Formalisms

Variety formalism

This automorphism property can be described in the language of universal algebra, viewing groups as a variety of algebras
View other such automorphism properties

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. For full proof, refer: Inner automorphisms are I-automorphisms in variety of groups

Relation with other properties

Stronger properties

Weaker properties

Class-preserving automorphism: For proof of the implication, refer Inner implies class-preserving and for proof of its strictness (i.e. the reverse implication being false) refer Class-preserving not implies inner.
Subgroup-conjugating automorphism
Extensible automorphism: For full proof, refer: Inner implies extensible
Pushforwardable automorphism
Monomial automorphism

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
View a complete list of group-closed automorphism properties

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. In fact, they form a normal subgroup.

Extensibility-stability

This function property is extensibility-stable, that is, given any embedding of groups, a function with the property in the smaller group can be lifted to a function with the property in the bigger group

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

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

For full proof, refer: Inner is extensibility-stable

Pushforwardability-stability

This function property is pushforwardability-stable, viz given any homomorphism of groups, a function with that property in the source group can be pushed forward to a function with the property in the target group

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

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

For full proof, refer: Inner is pushforwardability-stable

Template:QP-stable

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

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

For full proof, refer: Inner is quotient-pullbackability-stable

Template:Dirprodclosedap

If $G 1$ and $G 2$ are two groups, and $σ 1$ and $σ 2$ 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 $σ 1$ on the first coordinate and $σ 2$ on the second.

The idea is to take $g 1, g 2$ as conjugating candidates for $σ 1,σ 2$ . Then the element $(g 1, g 2)$ serves as a conjugating candidate for $\sigma_1 \times \sigma_2$ .

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN 0471433349, ^{More info}, Page 134 (formal definition, along with definition of the inner automorphism group)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 14 (definition introduced in paragraph)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 26 (Robinson uses the right action convention for inner automorphisms)
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 71 (formal definition)
Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 26 (formal definition, after the notion of conjugation by an element)
A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 175, Definition 3.2.9 (along with automorphism, formal definition)
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 90-91, (definition introduced in the context of Corollary 4.7(i))
Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 123
Topics in Algebra by I. N. Herstein, ^{More info}, Page 68 (definition introduced in paragraph)
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 195, Exercise 2(c) of Miscellaneous Problems (definition introduced in exercise)

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Definition links

Facts about Inner automorphismRDF feed

Defined in	DummitFoote (?, ?, ?) +, AlperinBell (?, ?, ?) +, RobinsonGT (?, ?, ?) +, RobinsonAA (?, ?, ?) +, Lang (?, ?, ?) +, Fraleigh (?, ?, ?) +, Hungerford (?, ?, ?) +, Gallian (?, ?, ?) +, Herstein (?, ?, ?) +, Artin (?, ?, ?) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, Mathworld (?, ?, ?) +, and Springer Online Reference Works (?, ?, ?) +
Defining ingredient	Conjugation +
Referenced in	DummitFoote (?, ?, ?) +, AlperinBell (?, ?, ?) +, RobinsonGT (?, ?, ?) +, RobinsonAA (?, ?, ?) +, Lang (?, ?, ?) +, Fraleigh (?, ?, ?) +, Hungerford (?, ?, ?) +, Gallian (?, ?, ?) +, Herstein (?, ?, ?) +, Artin (?, ?, ?) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, Mathworld (?, ?, ?) +, and Springer Online Reference Works (?, ?, ?) +
Stronger than	Class-preserving automorphism +, Subgroup-conjugating automorphism +, Extensible automorphism +, Pushforwardable automorphism +, and Monomial automorphism +
Weaker than	Central inner automorphism +, and Inner power automorphism +