Editing Group representation (section)

===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]].