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 gives rise to the concept of the outer automorphism group.
Definition
If  is a group and  is an element of  (alternatively, if  is a ring, and  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  (see also conjugacy class).
This function is an endomorphism of : for all x_1,x_2\in G,
\varphi_g(x_1 x_2) = g^{-1} x_1 x_2g = \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  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 conjugation gives a right action of  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  is a group, the inner automorphism group of  denoted .
is a normal subgroup of the full automorphism group  of .
The outer automorphism group,  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  are not inner.
Every non-inner automorphism yields a non-trivial element of , but different non-inner automorphisms may yield the same element of .
Saying that conjugation of  by  leaves  unchanged is equivalent to saying that  and  commute:
a^{-1}xa = x \iff ax = xa.
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  is inner if and only if it extends to every group containing .
By associating the element  with the inner automorphism  in  as above, one obtains an isomorphism between the quotient group  (where  is the center of ) and the inner automorphism group:
G/Z(G) \cong \operatorname{Inn}(G).
This is a consequence of the first isomorphism theorem, because  is precisely the set of those elements of  that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
Non-inner automorphisms of finite {{mvar|p}}-groups
A result of Wolfgang Gaschütz says that if  is a finite non-abelian -group, then  has an automorphism of -power order which is not inner.
It is an open problem whether every non-abelian -group  has an automorphism of order .
The latter question has positive answer whenever  has one of the following conditions:
is nilpotent of class 2
is a regular -group
is a powerful -group
The centralizer in , , of the center, , of the Frattini subgroup, , of , , is not equal to
Types of groups
The inner automorphism group of a group , , is trivial (i.e., consists only of the identity element) if and only if  is abelian.
The group  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  elements when  is not 2 or 6.
When , the symmetric group has a unique non-trivial class of outer automorphisms, and when , the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.
If the inner automorphism group of a perfect group  is simple, then  is called quasisimple.
Lie algebra case
An automorphism of a Lie algebra  is called an inner automorphism if it is of the form , where  is the adjoint map and  is an element of a Lie group whose Lie algebra is .
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  is the group of units of a ring, , then an inner automorphism on  can be extended to a mapping on the projective line over  by the group of units of the matrix ring, .
In particular, the inner automorphisms of the classical groups can be extended in that way.
References
Further reading
Category:Group theory Category:Group automorphisms
de:Automorphismus#Innere Automorphismen
