group. 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 . The natural action of group G1 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. Oligomorphic groups When a group G acts on a set S, the action may be extended naturally to the Cartesian product Sn of S, consisting of n-tuples of elements of S: the action of an element g on the n-tuple (s1, ..., sn) is given by g(s1, ..., sn) = (g(s1), ..., g(sn)). The group G is said to be oligomorphic if the action on Sn has only finitely many orbits for every positive integer n. (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 automorphisms in countably categorical theories. History The study of groups originally grew out of an understanding of permutation groups. Permutations had themselves been intensively studied by Lagrange in 1770 in his work on the algebraic solutions of polynomial equations. This subject flourished and by the mid 19th century a well-developed theory of permutation groups existed, codified by Camille Jordan in his book Traité des Substitutions et des Équations Algébriques of 1870. Jordan's book was, in turn, based on the papers that were left by Évariste Galois in 1832. When Cayley introduced the concept of an abstract group, it was not immediately clear whether or not this was a larger collection of objects than the known permutation groups (which had a definition different from the modern one). Cayley went on to prove that the two concepts were equivalent in Cayley's theorem. Another classical text containing several chapters on permutation groups is Burnside's Theory of Groups of Finite Order of 1911. The first half of the twentieth century was a fallow period in the study of group theory in general, but interest in permutation groups was revived in the 1950s by H. Wielandt whose German lecture notes were reprinted as Finite Permutation Groups in 1964. See also 2-transitive group Rank 3 permutation group Mathieu group Notes References Further reading Akos Seress. Permutation