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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 nonexamples). The first thing we shall prove is: All pgroups are nilpotent. Proof. Let G be a pgroup; 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 / nonexamples 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 qsubgroup N is … Continue reading
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Group Theory XI.1 (Solvable Groups)
This topic is rather detached from the rest, but it is sufficiently important that we can’t afford to drop it. Recall, that the finite simple groups form the building blocks of all finite groups. ( Of course we need to solve … Continue reading
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Group Theory X.4 (Optional: Burnside’s Problem)
We already know it’s easy to construct an infinite group where every element has finite order. For example, a product of infinite copies of C2. But that’s because we cheat by constructing infinitely many copies of finiteorder objects. So let’s constrain … Continue reading
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Group Theory X.3 (Word Problem for Groups)
Warning: this section has almost no proofs! The presentation of a group allows us to describe a group rather economically. For example, only 2 symbols and 3 relations are required to describe the nonabelian group of order 21. Thus there’s … Continue reading
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Group Theory X.2 (Presentation of a Group)
Let X be a set and F(X) be the free group on X. If Y is subset of F(X), we consider the normal subgroup N(Y) generated by Y. [ We’ve not mentioned this before, but the intersection of a collection … Continue reading
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Group Theory X.1 (Free Groups)
Consider a general group G with subset S. Without knowing anything else, what can we say about the generated subgroup ? If S has only one element, we simply get a cyclic subgroup. If S = {a, b}, things start … Continue reading
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