Inner automorphism
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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)
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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 of a group is termed an inner automorphism if there is an element in such that for all , .
Note that the choice of such that need not be unique. In fact, the possibilities for , for any , form a coset of the center of .
Convention
If the convention we choose is of left actions, then the inner automorphism is denoted as , and is termed the inner automorphism induced by (or conjugation by ). It is also sometimes denoted as .
If the convention is to make the group act on the right, the inner automorphism induced by is defined as , and is denoted as . Note that conjugation by in one convention equals conjugation by 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.
For more definitions of inner automorphism, check out nonstandard definitions of inner automorphism.
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 is denoted as . The image, which is the inner automorphism group, is thus .
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
Further information: extensible equals inner (specific proof), extensible automorphisms problem (more discussion)
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
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
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Inner power automorphism | inner automorphism that is also a power map |
Weaker properties
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. This follows from the fact that group acts as automorphisms by conjugation.
In fact, they form a normal subgroup of the automorphism group.
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 are groups and is an inner automorphism of , then there exists an inner automorphism of such that the restriction of to is .
The idea is to take any conjugating candidate for and consider the corresponding conjugation in the whole of .
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 is a homomorphism of groups, and is an inner automorphism of , then there exists an inner automorphism of such that .
The idea is to take any conjugating candidate for and define as conjugation by .
For full proof, refer: Inner is pushforwardability-stable
If is a surjective homomorphism of groups, and is an inner automorphism of , there exists an inner automorphism of such that .
The idea is to take any conjugating candidate for , pick any inverse image of via , and consider conjugation by that inverse element.
For full proof, refer: Inner is quotient-pullbackability-stable
If and are two groups, and and are inner automorphisms on and on respectively, then is an inner automorphism on . Here, is the automorphism of that acts as on the first coordinate and on the second.
The idea is to take as conjugating candidates for . Then the element serves as a conjugating candidate for .
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{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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