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 simple 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{align}

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

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


 * $$\varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = g^{-1} x_1 \left(g g^{-1}\right) x_2 g = \left(g^{-1} x_1 g\right)\left(g^{-1} x_2 g\right) = \varphi_g(x_1)\varphi_g(x_2),$$

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 $$\varphi_{g^{-1}}.$$ Thus, $$\varphi_g$$ is bijective, and so an isomorphism of $G$ with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.

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: $$\left(x^{g_1}\right)^{g_2} = x^{g_1g_2}$$ for all $$g_1, g_2 \in G.$$ This shows that right conjugation gives a right action of $G$ on itself.

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

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

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 $G$.

By associating the element $a ∈ G$ with the inner automorphism $f(x) = xa$ 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:
 * $$G\,/\,\mathrm{Z}(G) \cong \operatorname{Inn}(G).$$

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).

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 $G$, $CG$, of the center, $Z$, of the Frattini subgroup, $Φ$, of $G$, $CG ∘ Z ∘ Φ(G)$, 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 $Adg$, 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.