6: Permutation and Dihedral Groups
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We have already been introduced to two important classes of nonabelian groups: namely, the matrix groups \(GL(n,\mathbb{R})\) and \(SL(n,\mathbb{R})\) for \(n≥2\). We now consider a more general class of (mostly) nonabelian groups: permutation groups.
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6.1: Introduction to Permutation Groups
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In this section, we will introduce permutation groups and define permutation multiplication.
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6.2: Symmetric Groups
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In this section, we will discuss symmetric groups and cycle notation, as well as provide the definition and examples of disjoint cycles.
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6.3: Alternating Groups
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In this section, we will discuss alternating groups and corresponding theorems.
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6.4: Cayley's Theorem
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One might wonder how “common” permutation groups are in math. They are, it turns out, ubiquitous in abstract algebra: in fact, every group can be thought of as a group of permutations! We will prove this, but we first need to start with a lemma.
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6.5: Dihedral Groups
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Dihedral groups are groups of symmetries of regular n-gons. We will start with an example.
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6.6: Exercises
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This page contains the exercises for Chapter 6.