Monthly Archives: February 2011

Group Theory ∞ (Epilogue)

That concludes the end of the series of notes on Group Theory. Has it been successful? I don’t know, but I’m reasonably pleased with the way the notes turn out, except the fact that there’s a huge disparity between the … Continue reading

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Group Theory XII.5 (More Category Theory)

Since a category is really a bunch of abstract objects and arrows between them, we can reverse them by duality. Let C be a category. The opposite category Cop is the category such that: Ob(Cop) = Ob(C); for any objects … Continue reading

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Group Theory XII.4 (Category Theory: Functors)

In this section, we will explore further concepts in category theory. First, we shall talk about “maps” between categories. Let C, D be categories. A (covariant) functor (written as F : C → D) is a map F : Ob(C) → … Continue reading

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Group Theory XII.3 (More Universal Properties)

In this section, we shall get more practice with universal properties for various algebraic constructions. First, take the following categories: Set = category of sets, with morphisms = set functions; Grp = category of groups, with morphisms = group homomorphisms; … Continue reading

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Group Theory XII.2 (Category Theory)

Let’s take a closer look at the proofs and definition in the previous section. What concepts have we used? We have considered groups, homomorphisms between them, composition of homomorphisms, identity homomorphisms, isomorphisms and inverse homomorphisms. But the last two items … Continue reading

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Group Theory XII.1 (Introduction to Diagrams)

Given groups G and H, recall we have the product P = G × H and projection maps π1:P → G and π2:P → H. There’s nothing mysterious about the projection maps: these just take (g, h) to g and h respectively. But … Continue reading

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Group Theory XI.4 (Nilpotent Groups: Examples etc)

We will talk more about the properties of nilpotent groups, before narrowing down to the concrete examples (as well as non-examples). The first thing we shall prove is: All p-groups are nilpotent. Proof. Let G be a p-group; its centre Z(G) is … Continue reading

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Group Theory XI.3 (Nilpotent Groups)

Related to solvable groups is the concept of nilpotent groups. Let G be a group. We say G is nilpotent if there is a sequence of decreasing normal subgroups of G: such that G/Gn+1 commutes with Gn/Gn+1 for all n. Once again, … Continue reading

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Group Theory 0 (Preface + Table of Contents)

Welcome. In this series of notes, I’ll be covering some basic materials in Group Theory. Ok, truth is, even though I said “basic”, there’ll probably be enough materials to cover two semesters of undergraduate algebra. The aim of this set … Continue reading

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Group Theory XI.2 (Solvable Groups: Examples etc)

First, let us talk about some examples / non-examples of solvable groups. All abelian groups are solvable, since G(1) is trivial. Any group G of order pq for primes p < q is solvable; indeed the Sylow q-subgroup N is … Continue reading

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