(1432), respectively. The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). The only remaining symmetry is the identity (1)(2)(3)(4). This permutation group is abstractly known as the dihedral group of order 8. Group actions In the above example of the symmetry group of a square, the permutations "describe" the movement of the vertices of the square induced by the group of symmetries. It is common to say that these group elements are "acting" on the set of vertices of the square. This idea can be made precise by formally defining a group action. Let G be a group and M a nonempty set. An action of G on M is a function f: G × M → M such that f(1, x) = x, for all x in M (1 is the identity (neutral) element of the group G), and f(g, f(h, x)) = f(gh, x), for all g,h in G and all x in M. This pair of conditions can also be expressed as saying that the action induces a group homomorphism from G into Sym(M). Any such homomorphism is called a (permutation) representation of G on M. For any permutation group, the action that sends (g, x) → g(x) is called the natural action of G on M. This is the action that is assumed unless otherwise indicated. In the example of the symmetry group of the square, the group's action on the set of vertices is the natural action. However, this group also induces an action on the set of four triangles in the square, which are: t1 = 234, t2 = 134, t3 = 124 and t4 = 123. It also acts on the two diagonals: d1 = 13 and d2 = 24. Transitive actions The action of a group G on a set M is said to be transitive if, for every two elements s, t of M, there is some group element g such that g(s) = t. Equivalently, the set M forms a single orbit under the action of G. Of the examples above, the group {e, (1 2), (3 4), (1 2)(3 4)} of permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries