Inner automorphism

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.

Definition

If G is a group and g is an element of G (alternatively, if G is a ring, and g is a unit), then the function

$\begin{array}{cc}{\phi }_g:G& \to G\\ {\phi }_g(x)& :=g^{-1}xg\end{array}$

is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all $x_1,x_2\in G,$

${\phi }_g(x_1{x}_2)=g^{-1}{x}_1{x}_{2}g=g^{-1}{x}_1\left(g{g}^{-1}\right)x_{2}g=\left(g^{-1}{x}_{1}g\right)\left(g^{-1}{x}_{2}g\right)={\phi }_g(x_1){\phi }_g(x_2),$

where the second equality is given by the insertion of the identity between $x_1$ and $x_2.$ Furthermore, it has a left and right inverse, namely ${\phi }_{g^{-1}}.$ Thus, ${\phi }_g$ is both an monomorphism and epimorphism, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

File:Venn Diagram of Homomorphisms.jpg

General relationship between various group homomorphisms.

When discussing right conjugation, the expression $g^{-1}xg$ is often denoted exponentially by $x^g.$ This notation is used because composition of conjugations satisfies the identity: ${\left(x^{g_1}\right)}^{g_2}=x^{g_1{g}_2}$ for all $g_1,g_2\in G.$ This shows that right conjugation gives a right action of G on itself.

A common example is as follows:

File:Diagram of Inn(G) Example.jpg

Relationship of morphisms and elements

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

Let ${\phi }_g(x):=gx{g}^{-1}$ as defined above. This requires demonstrating that (1) ${\phi }_g$ is a homomorphism, (2) ${\phi }_g$ is also a bijection, (3) $\Phi$ is a homomorphism.

1. ${\phi }_g(x{x}^{\prime })=gx{x}^{\prime }{g}^{-1}=gx(g^{-1}g)x^{\prime }{g}^{-1}=(gx{g}^{-1})(g{x}^{\prime }{g}^{-1})={\phi }_g(x){\phi }_g(x^{\prime })$
2. The condition for bijectivity may be verified by simply presenting an inverse such that we can return to $x$ from $gx{g}^{-1}$. In this case it is conjugation by $g^{-1}$denoted as ${\phi }_{g^{-1}}$.
3. $\Phi (g{g}^{\prime })(x)=(g{g}^{\prime })x(g{g}^{\prime }{)}^{-1}$ and $\Phi (g)\circ \Phi (g^{\prime })(x)=\Phi (g)\circ (g^{\prime }xg{\text{'}}^{-1})=g{g}^{\prime }xg{\text{'}}^{-1}{g}^{-1}=(g{g}^{\prime })x(g{g}^{\prime }{)}^{-1}$

Inner and outer automorphism groups

The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(G).

Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The outer automorphism group, Out(G) is the quotient group

$\mathrm{Out}(G)=\mathrm{Aut}(G)/\mathrm{Inn}(G).$

The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).

Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:

$a^{-1}xa=x⟺xa=ax.$

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 G is inner if and only if it extends to every group containing G.

By associating the element a ∈ G with the inner automorphism f(x) = x^a in Inn(G) as above, one obtains an isomorphism between the quotient group G / Z(G) (where Z(G) is the center of G) and the inner automorphism group:

$G/\mathrm{Z}(G)\cong \mathrm{Inn}(G).$

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

Non-inner automorphisms of finite p-groups

A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner.

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

1. G is nilpotent of class 2
2. G is a regular p-group
3. G / Z(G) is a powerful p-group
4. The centralizer in G, C_G, of the center, Z, of the Frattini subgroup, Φ, of G, C_G ∘ Z ∘ Φ(G), is not equal to Φ(G)

Types of groups

The inner automorphism group of a group G, Inn(G), is trivial (i.e., consists only of the identity element) if and only if G is abelian.

The group Inn(G) 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 n elements when n is not 2 or 6. When n = 6, the symmetric group has a unique non-trivial class of non-inner automorphisms, and when n = 2, 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 G is simple, then G is called quasisimple.

Lie algebra case

An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it is of the form Ad_g, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is 𝔊. 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.

Extension

If G is the group of units of a ring, A, then an inner automorphism on G can be extended to a mapping on the projective line over A by the group of units of the matrix ring, M₂(A). In particular, the inner automorphisms of the classical groups can be extended in that way.