inner automorphism

Let $G$ be a group. For every $x\in G$, we define a mapping

$${\varphi }_x:G\to G,y\mapsto xy{x}^{-1},y\in G,$$

called conjugation by $x$. It is easy to show the conjugation map is in fact, a group automorphism.

An automorphism of $G$ that corresponds to conjugation by some $x\in G$ is called inner. An automorphism that isn’t inner is called an outer automorphism.

The composition operation gives the set of all automorphisms of $G$ the structure of a group, $\mathrm{Aut}(G)$. The inner automorphisms also form a group, $\mathrm{Inn}(G)$, which is a normal subgroup of $\mathrm{Aut}(G)$. Indeed, if ${\varphi }_x,x\in G$ is an inner automorphism and $\pi :G\to G$ an arbitrary automorphism, then

$$\pi \circ {\varphi }_x\circ {\pi }^{-1}={\varphi }_{\pi (x)}.$$

Let us also note that the mapping

$$x\mapsto {\varphi }_x,x\in G$$

is a surjective group homomorphism with kernel $\mathrm{Z}(G)$, the centre subgroup. Consequently, $\mathrm{Inn}(G)$ is naturally isomorphic to the quotient of $G/\mathrm{Z}(G)$.

Note: the above definitions and assertions hold, mutatis mutandi, if we define the conjugation action of $x\in G$ on $B$ to be the right action

$$y\mapsto x^{-1}yx,y\in G,$$

rather than the left action given above.

Title	inner automorphism
Canonical name	InnerAutomorphism
Date of creation	2013-03-22 12:49:53
Last modified on	2013-03-22 12:49:53
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	12
Author	rmilson (146)
Entry type	Definition
Classification	msc 20A05
Synonym	inner
Defines	conjugation
Defines	outer
Defines	outer automorphism
Defines	automorphism group