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3: Homomorphisms and Isomorphisms
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3.1: Groups of Small Order
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Let's start exploring groups in order of increasing, um, order. Before we do this, it will be helpful to introduce the notion of a group table (also known as a Cayley table).
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3.2: Definitions of Homomorphisms and Isomorphisms
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Intuitively, you can think of a homomorphism as a “structure-preserving” map: if you multiply and then apply homormorphism, you get the same result as when you first apply homomorphism and then multiply. Isomorphisms, then, are both structure-preserving and cardinality-preserving. Homomorphisms from a group G to itself are called endomorphisms, and isomorphisms from a group to itself are called automorphisms.
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3.3: Isomorphic Groups
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One of the key ideas we've discussed in determining whether binary structures are essentially “the same” or “different.” We approach this rigorously using the concept of isomorphic groups. Isomorphic groups have the same structure as far as algebraists are concerned.
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3.4: Exercises
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This page contains the exercises for Chapter 3.