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Group Theory 0 (Preface + Table of Contents) | Notes on Mathematics

https://limsup.wordpress.com/2011/02/24/group-theory-0-preface-table-of-contents

Just another WordPress.com site. Group Theory XI.2 (Solvable Groups: Examples etc). Group Theory XI.3 (Nilpotent Groups) →. Group Theory 0 (Preface Table of Contents). February 24, 2011. This set of notes is not intended to completely replace a textbook. Many of the proofs are omitted or contracted, there’re not enough exercises, and it’s certainly not healthy to stick to a single point-of-view. Group Theory I.1 (Introduction to Permutations). Group Theory I.2 (Notations for Permutations). Chapter IV tal...

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Group Theory IV.1 (Cosets) | Notes on Mathematics

https://limsup.wordpress.com/2011/02/16/group-theory-iv-1-cosets

Just another WordPress.com site. Group Theory III.4 (Properties of Subgroups). Group Theory IV.2 (Lagrange’s Theorem) →. Group Theory IV.1 (Cosets). February 16, 2011. Let’s compare subgroups with subsets; suppose. Is a subset of. Nothing more could be said. But now, if. Is a subgroup of. Something interesting happens: if. Is an element of. Then we define the (left) coset. It turns out that. Is also an element of. Thus we get the following picture:. Be a subgroup of. Be any elements of. Can be written as.

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Notes on Mathematics | Just another WordPress.com site | Page 2

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Just another WordPress.com site. Newer posts →. Group Theory XII.4 (Category Theory: Functors). February 25, 2011. 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. Written as F : C D) is a map F : Ob(C) Ob(D), and for any objects X, Y in C, a map F : Mor(X,Y) Mor(F(X), F(Y) , which respects the identity morphism and composition of morphisms, i.e. if X, Y, Z are objects in C, then:. X, Y) = Mor.

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Basic Algebra I.3 (Subrings and Ring Products) | Notes on Mathematics

https://limsup.wordpress.com/2011/03/03/basic-algebra-i-3-subrings-and-ring-products

Just another WordPress.com site. Basic Algebra I.2 (Examples Basic Properties of Rings). Basic Algebra I.4 (Ideals and Ring Quotients) →. Basic Algebra I.3 (Subrings and Ring Products). March 3, 2011. The concept of subrings follow naturally. Let R be a ring. A subring. Is a subset S of R such that:. S, ) is a subgroup. S contains 1;. For any a, b in S, ab is in S. We can actually simplify the above conditions to just the following:. Is non-empty and closed under subtraction, (. Must be a subgroup. Altho...

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Basic Algebra I.4 (Ideals and Ring Quotients) | Notes on Mathematics

https://limsup.wordpress.com/2011/03/03/basic-algebra-i-4-ring-quotients

Just another WordPress.com site. Basic Algebra I.3 (Subrings and Ring Products). Basic Algebra I.5 (Operations on Ideals) →. Basic Algebra I.4 (Ideals and Ring Quotients). March 3, 2011. Our first naïve attempt is to take a ring quotient. Is a subgroup of (. So we can denote every coset. Naturally we’ll want our 1. To be the identity element in. Which immediately gives us a problem: since 1 is in. And we have 1=0 in. 0 = 0 for any. This forces us to abandon our approach. Instead we look at. To be a subset.

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Notes on Mathematics | Just another WordPress.com site

Just another WordPress.com site. Basic Algebra I.5 (Operations on Ideals). March 3, 2011. We can perform the following operations to obtain new ideals:. Is an ideal of. Is an ideal of. Is an ideal of. Is the set of all finite sums of. Verifying that these sets are ideals is a rather routine process so we’ll leave it to the reader. Instead, we’ll spend some time explaining what these operations mean. The intersection. Needs no explanation: that’s the biggest ideal of. Is the smallest ideal which contains.

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