Abstract Nonsense

Crushing one theorem at a time

Review of Group Theory: Conjugacy Classes of The Product of Two Groups


Point of Post: In this post we show how one can completely classify the conjugacy classes of the product of two groups G and H given the conjugacy classes of each.

Motivation

It’s a natural question whether given two groups G and H and their set of conjugacy classes can we tell anything about the conjugacy classes of the product of the two groups. In fact, (as may or may not be obvious) there is an extremely easy way to find the set of all conjugacy classes of G\times H. Namely, if \mathcal{C}_G denotes the set of conjugacy classes of G and \mathcal{C}_H those of H then \mathcal{C}_{G\times H} is equal to \left\{A\times B:A\in\mathcal{C}_G\text{ and }B\in\mathcal{C}_H\right\}. Thus, in particular \#\left(\mathcal{C}_{G\times H}\right)=\#\left(\mathcal{C}_G\right)\#\left(\mathcal{C}_H\right).

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April 10, 2011 Posted by | Algebra, Group Theory | , , , , | 1 Comment

Review of Group Theory: Solvable Groups


Point of post: In this post we introduce the notion of solvable groups and derive some simple facts about them.

Motivation

Solvable groups are things that at first may seem useless. They have a somewhat complicated definition that seems almost completely arbitrary–in fact the opposite is true. The notion of solvable groups arose in the study of the solvability by radicals of certain polynomial equations, in particular in Galois theory. Specifically it was discovered that to every polynomial is associated a group (called the Galois group) and moreover it was discovered that the polynomial was solvable if and only if there existed a point after which a certain ‘normal chain of subgroups’ terminated (in the sense that after that point all the groups in the chain are trivial). With such an utterly fascinating result it is clear that the precise definition of this aforementioned condition, sufficient conditions for groups to have this condition, and properties that such groups have is absolutely important. This notion is the notion of solvability.

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March 11, 2011 Posted by | Algebra, Group Theory | , , , , , , , | 3 Comments

Review of Group Theory: The Commutator Subgroup and the Abelianization of a Group


Point of post: In this post we discuss the notion of the commutator subgroup of a group G and prove some basic, yet important, facts about it. We then discuss the abelianization of a group (the group mod the commutator subgroup).

Motivation

Very often in group theory we are dealt groups which, to our consternation, are not abelian. That said, being abelian and not abelian is often too black-and-white of a measure of commutativity. Thus, one may ask if there is some in-between quality that measures the ‘abelianess’ or lack thereof of the group. There is–this is the commutator subgroup. In essence the commutator subgroup is precisely such that when the ambient group is quotiented out by it everything of the form xyx^{-1}y^{-1} goes to the identity. This makes sense since in a perfect world, an abelian world that is, everything of this form would already be the identity. Thus, quotienting out by the subgroup generated by the set of all xyx^{-1}y^{-1} (i.e. the commutator subgroup) collapses all the defiant breakers of the golden abelian rule. Thus, in essence the bigger the commutator subgroup is the further from abelian the group is.

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February 27, 2011 Posted by | Algebra, Group Theory | , , , , , | 8 Comments

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


Point of post: In this post we give some alternate proofs of Sylow’s theorems which use a little more machinery then the “classic” ones.

 

Motivation

As remarked the Sylow Theorems are probably the most fundamentally important set of theorems in finite group theory. Consequently, any new proof verifying their validity is welcomed. In this post we give alternate proofs (in some sense) of the Sylow theorems than the “classic” proofs we have already given. They are less constructive and use more machinery. That said, they are very elegant and “cute”.

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January 11, 2011 Posted by | Algebra, Group Theory | , , | 1 Comment

Review of Group Theory: Semidirect Products (Pt. III The Dihedral Group)


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

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January 11, 2011 Posted by | Algebra, Group Theory | , , | Leave a comment

Review of Group Theory: Semidirect Products (Pt. II)


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

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January 11, 2011 Posted by | Algebra, Group Theory | , , | 1 Comment

Review of Group Theory: Semidirect Products (Pt. I)


Point of post: In this post we discuss the fruitful notion of semidirect products.

Motivation

Often the direct product of groups isn’t general enough to deal with most purposes. Thus, the idea of the semidirect product is to capture the spirit of what the direct product does but do so in a more general manner which with the suitable choice of homomorphis (see below) reduces to the case of the direct product anyways.

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January 11, 2011 Posted by | Algebra, Group Theory | , | 3 Comments

Review of Group Theory: Direct Product of Groups (Pt. II)


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

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January 10, 2011 Posted by | Algebra, Group Theory | , , | 3 Comments

Review of Group Theory: Direct Product of Groups


Point of post: In this post we cover the concept of direct product of finitely many groups

Motivation

By my hand being forced, I had to go quickly through all of this group theory. Consequently, things were done out of order and there is a particular dearth of examples in my posts. This post will be the first of two (the second will be on semidirect products) posts where we describe an interesting way to combine two (or more) groups to create a third. The concept is similar to most other mathematical structures. In essence, there is a canonical way to define a group structure on the Cartesian product of a finite number of groups. In fact, there is a canonical way to define the direct product and direct sum (the two coincide when only finitely many groups are involved) to arbitrary collections of groups. Sadly, time does not permit this level of detail

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January 10, 2011 Posted by | Algebra, Group Theory | , | 12 Comments