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Tag Archives: group theory
Free Groups and Tiling
Introduction Consider the following simple problem. Prove that the shape on the left cannot be completely tiled by 20 polygons of the types shown on the right. The solution is rather simple: colour the shape in the following manner. This … Continue reading
Posted in Uncategorized
Tagged combinatorics, free groups, group theory, groups, polyominoes, tiling, words
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Solving Permutation-Based Puzzles
Introduction In the previous article, we described the Schreier-Sims algorithm. Given a small subset which generates the permutation group G, the algorithm constructs a sequence such that for: we have a small generating set for each Specifically, via the Sims … Continue reading
Posted in Uncategorized
Tagged group actions, group theory, permutations, rubik's cube, schreier-sims, symmetric group
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Schreier-Sims Algorithm
Introduction Throughout this article, we let G be a subgroup of generated by a subset We wish to consider the following questions. Given A, how do we compute the order of G? How do we determine if an element lies in G? Assuming , how … Continue reading
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Tagged group actions, group theory, permutations, programming, rubik's cube, schreier-sims, symmetries
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Burnside’s Lemma and Polya Enumeration Theorem (1)
[ Note: this article assumes you know some rudimentary theory of group actions. ] Let’s consider the following combinatorial problem. Problem. ABC is a given equilateral triangle. We wish to colour each of the three vertices A, B and C by … Continue reading
Introduction to Ring Theory (6)
Let’s keep stock of what we’ve covered so far for ring theory, and compare it to the case of groups. There are loads of parallels between the two cases. G is a group R is a ring. Abelian groups. Commutative … 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
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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
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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
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Mathematics and Such 