Inner automorphism

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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 \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.
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 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

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

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Locally inner automorphism Resembles an inner automorphism on any finite subset (obvious) locally inner not implies inner |FULL LIST, MORE INFO
Class-preserving automorphism sends every element to within its conjugacy class inner implies class-preserving class-preserving not implies inner Hall-extensible automorphism, Locally inner automorphism|FULL LIST, MORE INFO
Subgroup-conjugating automorphism sends every subgroup to a conjugate subgroup inner implies subgroup-conjugating subgroup-conjugating not implies inner |FULL LIST, MORE INFO
Monomial automorphism can be expressed by a formula inner implies monomial monomial not implies inner Strong monomial automorphism|FULL LIST, MORE INFO
Strong monomial automorphism monomial and its inverse is monomial inner implies strong monomial strong monomial not implies inner |FULL LIST, MORE INFO
Normal automorphism sends every normal subgroup to itself inner implies normal normal not implies inner Class-preserving automorphism, Extended class-preserving automorphism, Hall-extensible automorphism, Locally inner automorphism, Rational class-preserving automorphism, Strong monomial automorphism, Subgroup-conjugating automorphism|FULL LIST, MORE INFO
Weakly normal automorphism sends every normal subgroup to a subgroup of itself (via normal) (via normal) Class-preserving automorphism, Extended class-preserving automorphism, Locally inner automorphism, Monomial automorphism, Normal automorphism, Rational class-preserving automorphism, Strong monomial automorphism|FULL LIST, MORE INFO
IA-automorphism induces identity automorphism on the abelianization inner implies IA IA not implies inner Automorphism that preserves conjugacy classes for a generating set, Class-preserving automorphism, Locally inner automorphism|FULL LIST, MORE INFO
Center-fixing automorphism fixes every element of the center inner implies center-fixing center-fixing not implies inner Class-preserving automorphism, Locally inner automorphism|FULL LIST, MORE INFO

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 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.

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 \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).

For full proof, refer: Inner is pushforwardability-stable

Template:QP-stable

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.

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

Template:Dirprodclosedap

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.

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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