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Category Archives: Notes
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
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Tagged advanced, conjugate, G-sets, group action, isotropy group, stabiliser group, transitive action
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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
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Tagged advanced, classification of finite simple groups, extension problem, finite groups, group theory, simple groups
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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
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Tagged advanced, factor through, group theory, homomorphism, isomorphism theorems, normal subgroups, universal properties
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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
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Tagged advanced, group quotients, group theory, groups, normal subgroups, subgroups
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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
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Tagged advanced, cosets, double cosets, group theory, groups, lagrange's theorem
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Casual Introduction to Group Theory (3)
Subgroups [ This article approximately corresponds to chapter III of the group theory blog. ] Let G be a group under operation *. If H is a subset of G, we wish to turn H into a group by inheriting the operation from G. Clearly, … Continue reading
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Tagged cyclic groups, generated groups, group theory applications, groups, intermediate, number theory, subgroups
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Casual Introduction to Group Theory (2)
Axioms of Group Theory [ This article approximately corresponds to chapter II of the earlier group theory blog. ] Group theory happens because mathematicians noticed that instead of looking at individual symmetries of an object, it’s far better to take … Continue reading
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Tagged abelian groups, axioms, canonical maps, group orders, group products, group theory, intermediate, isomorphisms
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Casual Introduction to Group Theory (1)
Introduction Last year, I created a blog which was supposed to introduce concepts to abstract algebra in a systematic manner. Though I was reasonably happy with the end result, I got the sneaky feeling upon completion that the end product … Continue reading
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Tagged 15 puzzle, abstract algebra, basic, casual, conjugate, group theory, order, permutations, rubik's cube, symmetries
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Combinatorial Game Theory XII
Lesson 12 Recall the following Domineering configuration in lesson 10: The above game has a nice theory behind it. Definition : For any game G, the game –G (called miny–G) is defined to be: The game +G (called tiny–G) is defined … Continue reading
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Tagged combinatorial game theory, computer science, domineering, game infinitesimals, impartial games, intermediate, minies, tinies, toads and frogs
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Combinatorial Game Theory XI
Lesson 11 In this lesson, we will cover more on canonical forms. First recall that for m↑ + (*n) with m > 0, this game is positive except when (m, n) = (1, 1). Let’s consider the canonical forms of … Continue reading
Mathematics and Such 