-
Recent Posts
Archives
- March 2023
- January 2023
- May 2020
- April 2020
- March 2020
- June 2018
- July 2016
- June 2016
- May 2016
- March 2015
- February 2015
- January 2015
- December 2014
- December 2013
- November 2013
- July 2013
- June 2013
- May 2013
- March 2013
- February 2013
- January 2013
- December 2012
- November 2012
- October 2012
- September 2012
- August 2012
- April 2012
- March 2012
- February 2012
- January 2012
- December 2011
- November 2011
- October 2011
Categories
Meta
Pages
Author Archives: limsup
Polynomial Multiplication, Karatsuba and Fast Fourier Transform
Let’s say you want to write a short program to multiply two linear functions f(x) = ax+b and g(x) = cx+d and compute the coefficients of the resulting product: You might think it’ll take 4 multiplications (for ac, ad, bc and bd) and 1 addition (for ad+bc), but there’s … Continue reading
Intermediate Group Theory (5)
Free Groups To motivate the concept of free groups, let’s consider some typical group G and elements a, b of G. Recall that , the subgroup generated by {a, b}, is defined to be the intersection of all subgroups of G containing a and b. Immediately, we see … Continue reading
Posted in Notes
Tagged advanced, free groups, generated groups, group theory, universal property
Leave a comment
Intermediate Group Theory (4)
Applications We’ll use the results that we obtained in the previous two posts to obtain some very nice results about finite groups. Example 1. A finite group G of order p2 is isomorphic to either Z/p2 or (Z/p) × (Z/p). In particular, it … Continue reading
Intermediate Group Theory (3)
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
Posted in Notes
Tagged advanced, automorphisms, conjugate, group actions, group theory, semidirect products
Leave a comment
Intermediate Group Theory (2)
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
Posted in Notes
Tagged advanced, cauchy's theorem, group actions, group theory, normaliser, sylow theorems
Leave a comment
Intermediate Group Theory (1)
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
Posted in Notes
Tagged advanced, conjugate, G-sets, group action, isotropy group, stabiliser group, transitive action
Leave a comment
Intermediate Group Theory (0)
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
Posted in Notes
Tagged advanced, classification of finite simple groups, extension problem, finite groups, group theory, simple groups
1 Comment
Casual Introduction to Group Theory (6)
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
Posted in Notes
Tagged advanced, factor through, group theory, homomorphism, isomorphism theorems, normal subgroups, universal properties
Leave a comment
Casual Introduction to Group Theory (5)
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
Posted in Notes
Tagged advanced, group quotients, group theory, groups, normal subgroups, subgroups
Leave a comment
Casual Introduction to Group Theory (4)
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
Posted in Notes
Tagged advanced, cosets, double cosets, group theory, groups, lagrange's theorem
Leave a comment
Mathematics and Such 