1. Even and Odd Permutations (10/2) Theorem. Every cycle, and hence every permutation, can be written as product of (usually non-disjoint) 2-cycles. Example. (1 2 3 4 5) = (1 5)(1 4)(1 3)(1 2). Note that this representation is not unique. For example, (1 2 3 4 5) = (2 1)(2 5)(2 4)(2 3)(1 4)(1 4) also. What is unique? Answer: Whether there are an odd number or an even number of 2-cycles. Theorem. If a permutation  can be written as an even number of 2-cycles, then every such representation of  will have an even number of 2-cycles. Likewise for odd. “Always Even or Always Odd”

2. More on Even and Odd Because of the preceding theorem, we can make the following definition: Definition. A permutation  is called even if it can be written as an even number of 2-cycles. Likewise for odd. We’ll call this the “type” of the permutation. Be sure to contrast this with the order of the permutation. They are different things!! Example: What type is (1 2 3 4 5)? What is its order? Give an example of an odd permutation of even order. Give an example of an even permutation of even order. Prove that there do not exist odd permutations of odd order!

3. Another Cool Result Theorem. Every group of permutations either consists entirely of even permutations, or it consists of exactly half even and half odd permutations. Examples: Check this with S 3 and S 4. Example. Thinking of D 4 as a subgroup of S 4 (with the vertices labeled 1 through 4), test out this theorem. Example. What about D 5 (as a subgroup of S 5 )? Theorem. The set of even permutations of any group of permutations G form a subgroup of G of order |G| or |G| / 2. Definition.The set of even permutations of S n is denoted A n and is called the alternating group on n elements.

4. Assignment for Friday Hand-in #2 is due on Monday. Read Chapter 5 from page 108 up to Example 8 (middle of page 111). Please do Exercises 8, 9, 10, 11, 15, 17, 22, 23, 24, 25 on pages 119-120.