outer automorphism group

The outer automorphism group of a group is the quotient (http://planetmath.org/QuotientGroup) of its automorphism group by its inner automorphism group:

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

There is some variance in terminology about “an outer automorphism.” Some authors define an outer automorphism as any automorphism $\varphi :G\to G$ which is not an inner automorphism. In this way an outer automorphism still permutes the group $G$. However, an equally common definition is to declare an outer automorphism as an element of $\mathrm{Out}(G)$ and consequently the elements are cosets of $\mathrm{Inn}(G)\varphi$, and not a map $\varphi :G\to G$. In this definition it is not generally possible to treat the element as a permutation of $G$. In particular, the outer automorphism group of a general group $G$ does not act on the group $G$ in a natural way. An exception is when $G$ is abelian so that $\mathrm{Inn}(G)=1$; thus, the elements of $\mathrm{Out}(G)$ are canonically identified with those of $\mathrm{Aut}(G)$ so we can speak of the action by outer automorphisms.

Title	outer automorphism group
Canonical name	OuterAutomorphismGroup
Date of creation	2013-03-22 14:01:26
Last modified on	2013-03-22 14:01:26
Owner	Thomas Heye (1234)
Last modified by	Thomas Heye (1234)
Numerical id	13
Author	Thomas Heye (1234)
Entry type	Definition
Classification	msc 20F28
Related topic	InnerAutomorphism
Defines	outer automorphism group