THE GROUP S n . BASIC PROPERTIES

The group Sn, n ≥ 1, is usually defined as the group of all permutations of the set Xn = {1, 2, . . . , n}. So, an element s ∈ Sn is a bijection (i.e., one-to-one map) from Xn to Xn : k 7→ s(k). The composition of two such map is denoted by s1 ◦ s2 : k 7→ s1 ( s2(k) ) . It is convenient to depict the permutation s as follows. Take two copy of Xn represented as column vectors and draw a system of n arrows joining element k of the first copy with the element s(k) of the second copy. To depict the composition s1◦s2 we use three copies of Xn. Then we join elements of the first copy with elements of the second one according to map s1 and elements of the second copy with elements of the third one according to map s2. Finally, we erase the second copy and “straighten” the broken lines joining elements of the first and third copies. We get the picture of s1 ◦ s2.