Inner automorphism

Template:Short description In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is defined as the outer automorphism group.

DefinitionEdit

If Template:Mvar is a group and Template:Mvar is an element of Template:Mvar (alternatively, if Template:Mvar is a ring, and Template:Mvar is a unit), then the function

<math>\begin{align}

  \varphi_g\colon G&\to G \\
  \varphi_g(x)&:= g^{-1}xg
\end{align}</math>

is called (right) conjugation by Template:Mvar (see also conjugacy class). This function is an endomorphism of Template:Mvar: for all <math>x_1,x_2\in G,</math>

<math>\varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = g^{-1} x_1 \left(g g^{-1}\right) x_2 g = \left(g^{-1} x_1 g\right)\left(g^{-1} x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2),</math>

where the second equality is given by the insertion of the identity between <math>x_1</math> and <math>x_2.</math> Furthermore, it has a left and right inverse, namely <math>\varphi_{g^{-1}}.</math> Thus, <math>\varphi_g</math> is both an monomorphism and epimorphism, and so an isomorphism of Template:Mvar with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.<ref>Template:Cite book</ref>

File:Venn Diagram of Homomorphisms.jpg

General relationship between various group homomorphisms.

When discussing right conjugation, the expression <math>g^{-1}xg</math> is often denoted exponentially by <math>x^g.</math> This notation is used because composition of conjugations satisfies the identity: <math>\left(x^{g_1}\right)^{g_2} = x^{g_1g_2}</math> for all <math>g_1, g_2 \in G.</math> This shows that right conjugation gives a right action of Template:Mvar on itself.

A common example is as follows:<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

File:Diagram of Inn(G) Example.jpg

Relationship of morphisms and elements

Describe a homomorphism <math>\Phi</math> for which the image, <math>\text{Im} (\Phi)</math>, is a normal subgroup of inner automorphisms of a group <math>G</math>; alternatively, describe a natural homomorphism of which the kernel of <math>\Phi</math> is the center of <math>G</math> (all <math>g \in G</math> for which conjugating by them returns the trivial automorphism), in other words, <math>\text{Ker} (\Phi) = \text{Z}(G)</math>. There is always a natural homomorphism <math>\Phi : G \to \text{Aut}(G) </math>, which associates to every <math>g \in G</math> an (inner) automorphism <math>\varphi_{g}</math> in <math>\text{Aut}(G)</math>. Put identically, <math>\Phi : g \mapsto \varphi_{g}</math>.

Let <math>\varphi_{g}(x) := gxg^{-1}</math> as defined above. This requires demonstrating that (1) <math>\varphi_{g}</math> is a homomorphism, (2) <math>\varphi_{g}</math> is also a bijection, (3) <math>\Phi</math> is a homomorphism.

<math>\varphi_{g}(xx')=gxx'g^{-1} =gx(g^{-1}g)x'g^{-1} = (gxg^{-1})(gx'g^{-1}) = \varphi_{g}(x)\varphi_{g}(x')</math>
The condition for bijectivity may be verified by simply presenting an inverse such that we can return to <math>x</math> from <math>gxg^{-1}</math>. In this case it is conjugation by <math>g^{-1}</math>denoted as <math>\varphi_{g^{-1}}</math>.
<math>\Phi(gg')(x)=(gg')x(gg')^{-1}</math> and <math>\Phi(g)\circ \Phi(g')(x)=\Phi(g) \circ (g'xg'^{-1}) = gg'xg'^{-1}g^{-1} = (gg')x(gg')^{-1}</math>

Inner and outer automorphism groupsEdit

The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of Template:Mvar is a group, the inner automorphism group of Template:Mvar denoted Template:Math.

Template:Math is a normal subgroup of the full automorphism group Template:Math of Template:Mvar. The outer automorphism group, Template:Math is the quotient group

<math>\operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Inn}(G).</math>

The outer automorphism group measures, in a sense, how many automorphisms of Template:Mvar are not inner. Every non-inner automorphism yields a non-trivial element of Template:Math, but different non-inner automorphisms may yield the same element of Template:Math.

Saying that conjugation of Template:Mvar by Template:Mvar leaves Template:Mvar unchanged is equivalent to saying that Template:Mvar and Template:Mvar commute:

Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).

An automorphism of a group Template:Mvar is inner if and only if it extends to every group containing Template:Mvar.<ref>Template:Citation</ref>

By associating the element Template:Math with the inner automorphism Template:Math in Template:Math as above, one obtains an isomorphism between the quotient group Template:Math (where Template:Math is the center of Template:Mvar) and the inner automorphism group:

<math>G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G).</math>

This is a consequence of the first isomorphism theorem, because Template:Math is precisely the set of those elements of Template:Mvar that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).

Non-inner automorphisms of finite Template:Mvar-groupsEdit

A result of Wolfgang Gaschütz says that if Template:Mvar is a finite non-abelian [[p-group|Template:Mvar-group]], then Template:Mvar has an automorphism of Template:Mvar-power order which is not inner.

It is an open problem whether every non-abelian Template:Mvar-group Template:Mvar has an automorphism of order Template:Mvar. The latter question has positive answer whenever Template:Mvar has one of the following conditions:

Template:Mvar is nilpotent of class 2
Template:Mvar is a [[regular p-group|regular Template:Mvar-group]]
Template:Math is a [[powerful p-group|powerful Template:Mvar-group]]
The centralizer in Template:Mvar, Template:Math, of the center, Template:Mvar, of the Frattini subgroup, Template:Math, of Template:Mvar, Template:Math, is not equal to Template:Math

Types of groupsEdit

The inner automorphism group of a group Template:Mvar, Template:Math, is trivial (i.e., consists only of the identity element) if and only if Template:Mvar is abelian.

The group Template:Math is cyclic only when it is trivial.

At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete. This is the case for all of the symmetric groups on Template:Mvar elements when Template:Mvar is not 2 or 6. When Template:Math, the symmetric group has a unique non-trivial class of non-inner automorphisms, and when Template:Math, the symmetric group, despite having no non-inner automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.

If the inner automorphism group of a perfect group Template:Mvar is simple, then Template:Mvar is called quasisimple.

Lie algebra caseEdit

An automorphism of a Lie algebra Template:Math is called an inner automorphism if it is of the form Template:Math, where Template:Math is the adjoint map and Template:Mvar is an element of a Lie group whose Lie algebra is Template:Math. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.

ExtensionEdit

If Template:Mvar is the group of units of a ring, Template:Mvar, then an inner automorphism on Template:Mvar can be extended to a mapping on the [[projective line over a ring|projective line over Template:Mvar]] by the group of units of the matrix ring, Template:Math. In particular, the inner automorphisms of the classical groups can be extended in that way.

ReferencesEdit

Template:Reflist