Inner automorphism

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

${\displaystyle {\begin{aligned}\varphi _{g}\colon G&\to G\\\varphi _{g}(x)&:=g^{-1}xg\end{aligned}}}$

is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all ${\displaystyle x_{1},x_{2}\in G,}$

${\displaystyle \varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=g^{-1}x_{1}\left(gg^{-1}\right)x_{2}g=\left(g^{-1}x_{1}g\right)\left(g^{-1}x_{2}g\right)=\varphi _{g}(x_{1})\varphi _{g}(x_{2}),}$

where the second equality is given by the insertion of the identity between ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}.}$ Furthermore, it has a left and right inverse, namely ${\displaystyle \varphi _{g^{-1}}.}$ Thus, ${\displaystyle \varphi _{g}}$ 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.^[1]

When discussing right conjugation, the expression ${\displaystyle g^{-1}xg}$ is often denoted exponentially by ${\displaystyle x^{g}.}$ This notation is used because composition of conjugations satisfies the identity: ${\displaystyle \left(x^{g_{1}}\right)^{g_{2}}=x^{g_{1}g_{2}}}$ for all ${\displaystyle g_{1},g_{2}\in G.}$ This shows that right conjugation gives a right action of G on itself.

A common example is as follows:^[2][3]

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

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

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

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

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

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:

${\displaystyle a^{-1}xa=x\iff xa=ax.}$

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.^[4]

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:

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

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).

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)

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.

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.

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.