Cycle Notation Cycle notation o Compute o Alternative

Cycle Notation

Cycle notation o Compute: o Alternative notation: (1 3)(2 5)(1 2 5 3 4) = (1 5)(3 4)

Products as disjoint cycles o (1 3)(2 5)(1 2 5 3 4) = (1 … = (1 5)(2)(3 4 … = (1 5)(2)(3 4) = (1 5)(3 4) Cycles not disjoint 1 --> 2 --> 5 5 --> 3 --> 1 2 --> 5 --> 2 3 --> 4 4 --> 1 --> 3 Eliminate unicycles : )

Thm 5. 1 Products of disjoint cycles o o Every permutation of a finite set can be written as a product of disjoint cycles. My proof: Let π be a permutation of a set A. Define a relation ~ on A as follows: a~b if πn(a) = b for some integer n > 0. Show ~ is an equivalence relation on A. So ~ partitions A into disjoint equivalence classes. The equivalence class of a can be written as the cycle (a π(a) π2(a)…πm-1(a)).

Thm 5. 2 o o Disjoint cycles commute. Example: Let =(124) = (35) Then =(124)(35) and =(35)(124) In array notation:

My Proof of 5. 2 o The Equivalence classes of the relation ~ do not depend on the order of listing.

Thm 5. 3 Order of a Permutation o o The order of a permutation written in disjoint cycles is the least common multiple of the lengths of the cycles. |(1 2 3 4)| = 4 |(5 6 7 8 9 10)| = 6 |(1 2 3 4)(5 6 7 8 9 10)| = lcm(4, 6) = 12 |(1 2 3)(3 4 5)| = |(1 2 3 4 5)| = 5

Thm 5. 4 Products of 2 -cycles o Every permutation in Sn for n ≥ 1 can be written as the product of 2 -cycles.