Cycle Notation

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

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

Thm 5.1 Products of disjoint cycles  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 π n (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) π 2 (a)…π m-1 (a)).

Thm 5.2  Disjoint cycles commute.  Example: Let  =(124)  = (35) Then  =(124)(35) and  =(35)(124) In array notation:

My Proof of 5.2  The Equivalence classes of the relation ~ do not depend on the order of listing.

Thm 5.3 Order of a Permutation  The order of a permutation written in disjoint cycles is the least common multiple of the lengths of the cycles.  |( )| = 4 |( )| = 6 |( )( )| = lcm(4,6) = 12 |(1 2 3)(3 4 5)| = |( )| = 5

Thm 5.4 Products of 2-cycles  Every permutation in S n for n ≥ 1 can be written as the product of 2-cycles.