Abstract Nonsense

Crushing one theorem at a time

Sylow’s Theorems Revisited


Point of Post: In this post we give a more refined proof of Sylow’s Theorems.

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Motivation

On this blog I have given a proof of Sylow’s theorems and an alternate proof of Sylow’s first theorem. A year or so later, with more time, and more finesse I’d like to give the simplest, most coherent proof of Sylow’s theorems.

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September 20, 2011 Posted by | Algebra, Group Theory | , , , , , , , | 4 Comments

Actions by p-Groups (Pt. II)


Point of Post: This is a continuation of this post.

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September 15, 2011 Posted by | Algebra, Group Theory | , , , , , , , , , , | 1 Comment

Actions by p-Groups (Pt. I)


Point of Post: In this post we discuss the theory of p-groups acting on sets, and some of its ramifications.

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Motivation

I have previously discussed group actions, but being a rush to discuss group theory I skirted over some of the beautiful theory. So, I’d like to take some time to discuss one of the prettier and more powerful branches of the theory, namely when we restrict our attention to group actions by p-groups. Not only will we be able to say some prove some fairly substantive  theorems about p-group actions explicitly, but will be able to prove some very neat things in more general group theory and in number theory. The interesting fact about the theory we will discuss is that at the root of everything is a ‘fundamental theorem’ whose presence (being the theorem in the case of a particular group action) is the proof that every p-group has a non-trivial center.  Namely, we were able to conclude that since the cardinality of any conjugacy class must divide the order of the group, that they must be divisible by p. From this and the fact that the sum of the cardinalities of all the distinct conjugacy classes must sum to the order of the group (which is divisible by p) that the sum of all the one point conjugacy classes must have cardinality divisible by p. Well, the generalization of this idea (which, as I’m sure is pretty clear, can be restated for an arbitrary action with conjugacy class replaced by orbit) will be the main tool I mentioned from which all our other theorems are (not always straight-forward) consequences.

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September 15, 2011 Posted by | Algebra, Group Theory | , , , , , , , , , , | 2 Comments

Representation Theory of Semidirect Products: The Preliminaries (Pt. III)


Point of Post: This is a continuation of this post.

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May 8, 2011 Posted by | Algebra, Representation Theory | , , , , , | Leave a comment

Review of Group Theory: Alternate Proof to the Sylow Theorems


Point of post: This is a continuation of this post.

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January 12, 2011 Posted by | Algebra, Group Theory | , , | Leave a comment

Review of Group Theory: Alternate Proof of the Sylow Theorems (Pt. II)


Point of post: This is a continuation of this post.

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January 12, 2011 Posted by | Algebra, Group Theory | , , | 1 Comment

Review of Group Theory: Group Actions (Pt. III G-Space Isomorphisms and the Fundamental Theorem of G-Spaces)


Point of post: In this post we describe what it means for two G-spaces to be isomorphic and describe the Fundamental Theorem of G-Spaces.

 

Motivation

Just as with all other structures it’s fruitful to define the maps between G-spaces that preserve the structure we’re interested in. It’s intuitively clear that for G-spaces this kind of map should preserve the action, in the sense that if we act on an element and then map it over we should get the same result if we map it over and then act on it.

After we have rigorously defined the notion of G-space isomorphism we describe the Fundamental Theorem of G-spaces.

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January 5, 2011 Posted by | Algebra, Group Theory | , , , | 1 Comment

Review of Group Theory: Group Actions (Pt. II Orbits and the Orbit Decomposition Theorem)


Point of post: In this post we discuss more in-depth the concept of the orbits of a G-action and show how it carves up a group into a partition, which for finite groups gives us a lot of information.

Motivation

In prior posts we’ve discussed Lagrange’s theorem and some of the profound consequences it can have on the study of finite groups. Look closely though and one will see that Lagrange’s theorem was really just a consequence of group actions. Namely, we had G was a finite group and H\leqslant G then H acted naturally on G by h\cdot g=hg. One can quick check that the orbits of these actions are the right cosets. The reason why this action was powerful was the way in which it carved up G into disjoint subsets, which we could exploit to tell us interesting things combinatorially/number theoretically about the order of G. It turns out that this wasn’t a coincidence. In particular, every G-action on a set S carves S up in a particularly nice way. This post will be devoted to studying in which way precisely this is.

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January 5, 2011 Posted by | Algebra, Group Theory, Uncategorized | , , , , , | 14 Comments

Review of Group Theory: Group Actions (Pt. I Definitions and a Sharpening of Cayley’s Theorem cont.)


Point of post: This post is a continuation of this one.

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January 4, 2011 Posted by | Algebra, Group Theory | , , , | 4 Comments

Review of Group Theorem: Group Actions (Pt. I Definitions and a Sharpening of Cayley’s Theorem)


Point of post: In this post we discuss the ideas of group actions and the counting arguments one can use them for.

Motivation

We now move into one of the most beautiful subjects of finite group theory, the theory of group actions. Group actions occur everywhere in mathematics, most of the time without us even taking notice. Intuitively a group acts on a set by moving it’s members around in a specific way. For example, the permutation group S_3 acts on the set of labeled vertices of a triangle by moving them into a different configuration.

We shall see that group actions enable us to tell a lot about the structure of the underlying groups. In particular, it will give us the class equation.

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January 4, 2011 Posted by | Algebra, Group Theory | , , , , | 11 Comments

   

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