Editing Permutation group (section)

== Oligomorphic groups ==

When a group ''G'' acts on a [[set (mathematics)|set]] ''S'', the action may be extended naturally to the [[Cartesian product]] ''S<sup>n</sup>'' of ''S'', consisting of ''n''-tuples of elements of ''S'': the action of an element ''g'' on the ''n''-tuple (''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub>) is given by
: ''g''(''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub>) = (''g''(''s''<sub>1</sub>), ..., ''g''(''s''<sub>''n''</sub>)).

The group ''G'' is said to be ''oligomorphic'' if the action on ''S<sup>n</sup>'' has only finitely many orbits for every positive integer ''n''.<ref>{{cite book | last=Cameron | first=Peter J. | author-link=Peter Cameron (mathematician) | title=Oligomorphic permutation groups | series=London Mathematical Society Lecture Note Series | volume=152 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1990 | isbn=0-521-38836-8 | zbl=0813.20002 }}</ref><ref>[http://www.newton.ac.uk/files/preprints/ni08029.pdf Oligomorphic permutation groups] - Isaac Newton Institute preprint, Peter J. Cameron</ref>  (This is automatic if ''S'' is finite, so the term is typically of interest when ''S'' is infinite.)

The interest in oligomorphic groups is partly based on their application to [[model theory]], for example when considering [[automorphism]]s in [[countably categorical theory|countably categorical theories]].<ref>{{cite book | zbl=0916.20002 | last1=Bhattacharjee | first1=Meenaxi | last2=Macpherson |first2=Dugald | last3=Möller | first3=Rögnvaldur G. | last4=Neumann | first4=Peter M. | title=Notes on infinite permutation groups | series=Lecture Notes in Mathematics | volume=1698 | location=Berlin | publisher=[[Springer-Verlag]] | year=1998 | isbn=3-540-64965-4 | page=83 }}</ref>