View source for Inner automorphism

''of a [[Group|group]] $G$''

An [[automorphism]] $\phi$ such that
$$
\phi(x) = g^{-1} x g
$$

for a certain fixed element $g \in G$. The set of all inner automorphisms of $G$ forms a normal subgroup in the group of all automorphisms of $G$; this subgroup is isomorphic to $G / Z(G)$, where $Z(G)$ is the centre of $G$ (cf. [[Centre of a group|Centre of a group]]). Automorphisms that are not inner are called outer automorphisms.

Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.



====Comments====
Let $\mathfrak{g}$ be a [[Lie algebra]] and $x \in \mathfrak{g}$ an element of $\mathfrak{g}$ for which $\mathrm{ad}(x) : y \mapsto [x,y]$ is nilpotent. Then
$$
\exp(\mathrm{ad}(x)) = \mathrm{id} + \mathrm{ad}(x) + \frac{1}{2!}\mathrm{ad}(x)^2 + \cdots
$$
defines an automorphism of $\mathfrak{g}$. Such an automorphism is called an inner automorphism of $\mathfrak{g}$. More generally, the elements in the group $\mathrm{Int}(\mathfrak{g})$ generated by them are called inner automorphisms. It is a normal subgroup of $\mathrm{Aut}(\mathfrak{g})$.

If $G$ is a real or complex [[Lie group]] with semi-simple Lie algebra, then the inner automorphisms constitute precisely the identity component of the group $\mathrm{Aut}(\mathfrak{g})$ of automorphisms of $\mathfrak{g}$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,   "The theory of groups" , Macmillan  (1959)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer  (1972)  pp. §5.4</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.-P. Serre,   "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR>
</table>