Even and Odd Permutations (10/2)
• Theorem. Every cycle, and hence every permutation, can
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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”
More on Even and Odd
• Because of the preceding theorem, we can make the
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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!
Another Cool Result
• Theorem. Every group of permutations either consists
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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 An
and is called the alternating group on n elements.
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.