Editing Character table (section)

== Outer automorphisms ==
The [[outer automorphism]] group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism <math>g \mapsto g^{-1}</math>, which is non-trivial except for [[elementary abelian group|elementary abelian 2-groups]], and outer because abelian groups are precisely those for which conjugation ([[inner automorphism]]s) acts trivially. In the example of <math>C_3</math> above, this map sends <math>u \mapsto u^2, u^2 \mapsto u,</math> and accordingly switches <math>\chi_1</math> and <math>\chi_2</math> (switching their values of <math>\omega</math> and <math>\omega^2</math>). Note that this particular [[automorphism]] (negative in abelian groups) agrees with complex conjugation.

Formally, if <math>\phi\colon G \to G</math> is an automorphism of ''G'' and <math>\rho \colon G \to \operatorname{GL}</math> is a representation, then <math>\rho^\phi := g \mapsto \rho(\phi(g))</math> is a representation. If <math>\phi = \phi_a</math> is an [[inner automorphism]] (conjugation by some element ''a''), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism group <math>\mathrm{Aut}</math> descends to the [[quotient group|quotient]] <math>\mathrm{Out}</math>.

This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.