-
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
Monthly Archives: October 2012
Intermediate Group Theory (6)
In this post, we’ll only focus on additive abelian groups. By additive, we mean the underlying group operation is denoted by +. The identity and inverse of x are denoted by 0 and –x respectively. Similarly, 2x+3y refers to x+x+y+y+y. Etc … Continue reading
Posted in Notes
Tagged abelian groups, advanced, direct products, direct sums, free groups, generated groups, universal properties
Leave a comment
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
Mathematics and Such 