Editing Group representation (section)

==Generalizations==

===Set-theoretical representations===
A ''set-theoretic representation'' (also known as a group action or ''permutation representation'') of a [[group (mathematics)|group]] ''G'' on a [[Set (mathematics)|set]] ''X'' is given by a [[function (mathematics)|function]] ρ : ''G'' → ''X''<sup>''X''</sup>, the set of functions from ''X'' to ''X'', such that for all ''g''<sub>1</sub>, ''g''<sub>2</sub> in ''G'' and all ''x'' in ''X'':

:<math>\rho(1)[x] = x</math>
:<math>\rho(g_1 g_2)[x]=\rho(g_1)[\rho(g_2)[x]],</math>

where <math>1</math> is the identity element of ''G''. This condition and the axioms for a group imply that ρ(''g'') is a [[bijection]] (or [[permutation]]) for all ''g'' in ''G''. Thus we may equivalently define a permutation representation to be a [[group homomorphism]] from G to the [[symmetric group]] S<sub>''X''</sub> of ''X''.

For more information on this topic see the article on [[Group action (mathematics)|group action]].

===Representations in other categories===
Every group ''G'' can be viewed as a [[category (mathematics)|category]] with a single object; [[morphism]]s in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a [[functor]] from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the [[automorphism group]] of ''X''.

In the case where ''C'' is '''Vect'''<sub>''K''</sub>, the [[category of vector spaces]] over a field ''K'', this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the [[category of sets]].

When ''C'' is '''Ab''', the [[category of abelian groups]], the objects obtained are called [[G-module|''G''-modules]].

For another example consider the [[category of topological spaces]], '''Top'''. Representations in '''Top''' are homomorphisms from ''G'' to the [[homeomorphism]] group of a topological space ''X''.

Two types of representations closely related to linear representations are:
*[[projective representation]]s: in the category of [[projective space]]s. These can be described as "linear representations [[up to]] scalar transformations". 
*[[affine representation]]s: in the category of [[affine space]]s. For example, the [[Euclidean group]] acts affinely upon [[Euclidean space]].