ABSTRACT
The symmetric group Sn is the group of bijections of {1, . . . , n} to itself, also called permutations of n things. A standard notation for the permutation that sends i −→ `i is(
1 2 3 . . . n `1 `2 `3 . . . `n
) Under composition of mappings, the permutations of {1, . . . , n} is a group. The fixed points of a permutation f are the elements i ∈ {1, 2, . . . , n} such that f(i) = i. A k-cycle is a permutation of the form
f(`1) = `2 f(`2) = `3 . . . f(`k−1) = `k and f(`k) = `1
for distinct `1, . . . , `k among {1, . . . , n}, and f(i) = i for i not among the `j . There is standard notation for this cycle:
(`1 `2 `3 . . . `k)
Note that the same cycle can be written several ways, by cyclically permuting the `j : for example, it also can be written as
(`2 `3 . . . `k `1) or (`3 `4 . . . `k `1 `2)
Two cycles are disjoint when the respective sets of indices properly moved are disjoint. That is, cycles (`1 `2 `3 . . . `k) and (`′1 `
′ 3 . . . `
