Day 4

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Section Day 4

This is an outline of the topics we covered in the fourth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.

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Handout Thursday 5/28

Objectives: Advanced Learning Outcomes

During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.

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Use cycle notation to compose permutations
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Write permutations as products of disjoint cycles
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State the following mathematical results: Theorem 5.2 (Disjoint Cycles Commute), Theorem 5.3 (Order of a Permutation)
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State and instantiate the definitions of: even permutation, odd permutation
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State the following mathematical result: Theorem 5.5 (Always Even or Always Odd)
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Algebraist of the Day.

Ruth Moufang, 1905-1977

Studied projective geometry, mixing geometry and algebra.
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Worked in industry during Nazi Germany (possibly the first woman to work in industry with a doctorate)
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Returned to teaching in Frankfurt after the war but never published again
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Reminders/Announcements.

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Example 37. Cycle Notation.

Consider \(\alpha=\begin{bmatrix} 1 \amp 2 \amp 3 \amp 4 \amp 5 \\ 5 \amp 4 \amp 1 \amp 2 \amp 3 \end{bmatrix}\)

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Theorem 38. Every Permutation is a Product of Disjoint Cycles.

Every permutation of a finite set can be written as a product of disjoint cycles.

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Theorem 39. Disjoint Cycles Commute.

If \(\alpha=(a_1\;\dots \; a_m)\) and \(\beta=(b_1\;\dots \; b_n)\) are disjoint cycles, then \(\alpha\beta=\beta\alpha\text{.}\)

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Proof Idea:

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Theorem 40. Order of a Permutation.

The order of a permutation is the least common multiple of the lengths of the cycles in its disjoint cycle representation.

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Proof.

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Theorem 41. Every Permutation is a Product of Transpositions.

Every permutation in \(S_n\) for \(n\gt 1\) is a product of 2-cycles.

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Proof Idea:

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Proposition 42. Canonical Generators of \(S_n\).

The symmetric group \(S_n\) for \(n\geq 3\) has three canonical generating sets.

The set of all transpositions including 1: \(\{(1\;2), (1\;3), \ldots, (1\;n)\}\)
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The set of all adjacent transpositions: \(\{(1\;2),(2\;3), \ldots, (n-1\;n)\}\)
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An \(n\)-cycle and a well-chosen transposition: \(\{(1\;2\;\dots\; n), (1\;2)\}\)
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Theorem 43. Always Even or Always Odd.

If a permutation \(\alpha\in S_n\) can be written as a product of an even (odd) number of 2-cycles, then every decomposition of \(\alpha\) into a product of 2-cycles has an even (odd) number of 2-cycles.

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A permutation is called even if it can be written as a product of an even number of 2-cycles, and odd if it can be written as a product of an odd number of 2-cycles.

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