Editing Permutation group (section)

== Isomorphisms of permutation groups ==

If ''G'' and ''H'' are two permutation groups on sets ''X'' and ''Y'' with actions ''f''<sub>1</sub> and ''f''<sub>2</sub> respectively, then we say that ''G'' and ''H'' are ''permutation isomorphic'' (or ''[[isomorphism|isomorphic]] as permutation groups'') if there exists a [[Bijection|bijective map]] {{nowrap|''λ'' : ''X'' → ''Y''}} and a [[group isomorphism]] {{nowrap|''ψ'' : ''G'' → ''H''}} such that
: ''λ''(''f''<sub>1</sub>(''g'', ''x'')) = ''f''<sub>2</sub>(''ψ''(''g''), ''λ''(''x'')) for all ''g'' in ''G'' and ''x'' in ''X''.<ref>{{harvnb|Dixon|Mortimer|1996|p=17}}</ref>

If {{nowrap|1=''X'' = ''Y''}} this is equivalent to ''G'' and ''H'' being conjugate as subgroups of Sym(''X'').<ref>{{harvnb|Dixon|Mortimer|1996|loc=p. 18}}</ref> The special case where {{nowrap|1=''G'' = ''H''}} and ''ψ'' is the [[identity map]] gives rise to the concept of ''equivalent actions'' of a group.<ref>{{harvnb|Cameron|1994|loc=p. 228}}</ref>

In the example of the symmetries of a square given above, the natural action on the set {1,2,3,4} is equivalent to the action on the triangles. The bijection ''λ'' between the sets is given by {{nowrap|''i'' ↦ ''t''<sub>''i''</sub>}}. The natural action of group ''G''<sub>1</sub> above and its action on itself (via left multiplication) are not equivalent as the natural action has fixed points and the second action does not.