The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). ![]() The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. Let the vertices of a square be labeled 1, 2, 3 and 4 (counterclockwise around the square starting with 1 in the top left corner). This permutation group is, as an abstract group, the Klein group V 4.Īs another example consider the group of symmetries of a square. G 1 forms a group, since aa = bb = e, ba = ab, and abab = e. ![]() This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4.Like the previous one, but exchanging 3 and 4, and fixing the others.This permutation interchanges 1 and 2, and fixes 3 and 4.This is the identity, the trivial permutation which fixes each element.The term permutation group thus means a subgroup of the symmetric group. The group of all permutations of a set M is the symmetric group of M, often written as Sym( M). ![]() In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).
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