indicate multiplication (the dots of the previous example were added for emphasis, so would simply be written as ). Neutral element and inverses The identity permutation, which maps every element of the set to itself, is the neutral element for this product. In two-line notation, the identity is In cycle notation, e = (1)(2)(3)...(n) which by convention is also denoted by just (1) or even (). Since bijections have inverses, so do permutations, and the inverse σ−1 of σ is again a permutation. Explicitly, whenever σ(x)=y one also has σ−1(y)=x. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance To obtain the inverse of a single cycle, we reverse the order of its elements. Thus, To obtain the inverse of a product of cycles, we first reverse the order of the cycles, and then we take the inverse of each as above. Thus, Having an associative product, an identity element, and inverses for all its elements, makes the set of all permutations of M into a group, Sym(M); a permutation group. Examples Consider the following set G1 of permutations of the set M = {1, 2, 3, 4}: e = (1)(2)(3)(4) = (1) This is the identity, the trivial permutation which fixes each element. a = (1 2)(3)(4) = (1 2) This permutation interchanges 1 and 2, and fixes 3 and 4. b = (1)(2)(3 4) = (3 4) Like the previous one, but exchanging 3 and 4, and fixing the others. ab = (1 2)(3 4) This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4. G1 forms a group, since aa = bb = e, ba = ab, and abab = e. This permutation group is isomorphic, as an abstract group, to the Klein group V4. As another example consider the group of symmetries of a square. 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). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The 180° and 270° rotations are given by (13)(24) and