Cycle Notation

Cycle notation

Compute:

Alternative notation:

(1 3)(2 5)(1 2 5 3 4) = (1 5)(3 4)€

1 2 3 4 5

3 5 1 4 2

⎡

⎣ ⎢

⎤

⎦ ⎥

1 2 3 4 5

2 5 4 1 3

⎡

⎣ ⎢

⎤

⎦ ⎥=

€

1 2 3 4 5

5 2 4 3 1

⎡

⎣ ⎢

⎤

⎦ ⎥

Products as disjoint cycles (1 3)(2 5)(1 2 5 3 4)

= (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 --> 5

5 --> 3 --> 3 --> 1

2 --> 5 --> 2 --> 2

3 --> 4 --> 4 --> 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 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

Disjoint cycles commute. Example: Let =(124) = (35)

Then =(124)(35)

and =(35)(124)

In array notation:

€

=1 2 3 4 5

2 4 5 1 3

⎡

⎣ ⎢

⎤

⎦ ⎥= βα

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.

|(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

Every permutation in Sn for n ≥ 1 can be written as the product of 2-cycles.