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Even and Odd Permutations (10/2)

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).

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Even and Odd Permutations (10/2)

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  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 evenif 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 S3 and S4. • Example. Thinking of D4 as a subgroup of S4 (with the vertices labeled 1 through 4), test out this theorem. • Example. What about D5 (as a subgroup of S5)? • 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 Sn is denoted Anand 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.

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