Tag Archives: advanced

Intermediate Group Theory (3)

Posted on October 3, 2012 by limsup

Automorphisms and Conjugations of G We’ve seen how groups can act on sets via bijections. If the underlying set were endowed with a group structure, we can restrict our attention to bijections which preserve the group operation. Definition. An automorphism of … Continue reading →

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Intermediate Group Theory (2)

Posted on October 2, 2012 by limsup

This is a continuation from the previous post. Let G act on set X, but now we assume that both G and X are finite. Since X is a disjoint union of transitive G-sets, and each transitive G-set is isomorphic to G/H for some subgroup H ≤ G, it follows that … Continue reading →

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Intermediate Group Theory (1)

Posted on October 1, 2012 by limsup

Given a group G, we wish to find out more about its properties. Questions include: what subgroups does it have? And normal subgroups? How many elements of order m does it have (where m must divide the order of G if the latter is finite)? … Continue reading →

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Intermediate Group Theory (0)

Posted on September 30, 2012 by limsup

Let’s take stock of what we know about group theory so far in the first series. We defined a group, which is a set endowed with a binary operation satisfying 3 properties. For each group, we considered subsets which could … Continue reading →

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Casual Introduction to Group Theory (6)

Posted on September 28, 2012 by limsup

Homomorphisms [ This post roughly corresponds to Chapter VI of the old blog. ] For sets, one considers functions f : S → T between them. For groups, one would like to consider only actions which respect the group operation. Definition.  Let G and … Continue reading →

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Casual Introduction to Group Theory (5)

Posted on September 26, 2012 by limsup

Normal Subgroups and Group Quotients [ This corresponds to approximately chapter V of the old blog. ] We’ve already seen that if H ≤ G is a subgroup, then G is a disjoint union of (left) cosets of H in G. We’d like to use the set … Continue reading →

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Casual Introduction to Group Theory (4)

Posted on September 25, 2012 by limsup

Cosets and Lagrange’s Theorem [ This post approximately corresponds to chapter IV from the old group theory blog. ] The main theorem in this post is Lagrange’s theorem: if H ≤ G is a subgroup then |H| divides |G|. But first, let’s consider … Continue reading →

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Power Series and Generating Functions (IV) – Exponential Generating Functions

Posted on March 26, 2012 by limsup

Note: this article is noticeably more difficult than the previous instalments. The reader is advised to be completely comfortable with generating functions before proceeding. We’ve already seen how generating functions can be used to solve some combinatorial problems. The nice … Continue reading →

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Linear Algebra: Inner Products

Posted on December 2, 2011 by limsup

[ Background required: basic knowledge of linear algebra, e.g. the previous post. Updated on 6 Dec 2011: added graphs in Application 2, courtesy of wolframalpha.] Those of you who already know inner products may roll your eyes at this point, … Continue reading →

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Number Theory and Calculus/Analysis

Posted on November 1, 2011 by limsup

Background required: modular arithmetic, calculus. Once in a while, I’ll post something which offers a glimpse into more advanced mathematics. Here’s one. Example 1 For starters, we know from basic algebra that . Let’s see if there’s a corresponding result … Continue reading →

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