Abstract Nonsense

Crushing one theorem at a time

Q and Q^2 are not Isomorphic as Groups


Point of Post: In this post we prove that \mathbb{Q}\not\cong\mathbb{Q}^2 as abelian groups, and discuss a generalization.

\text{ }

Motivation

\text{ }

As I was trying to go to sleep last night, I was pondering the contents of my post the other day. In particular, I was thinking about the statement that I made that “one should always ask the obvious questions”, whatever that’s supposed to mean. It hit me then, that while there are many ‘obvious’ questions that could follow from the fact that \mathbb{R}\cong\mathbb{C}\cong\mathbb{R}^2 as abelian groups, I didn’t ask perhaps the most obvious one. Namely, is \mathbb{Q}\cong\mathbb{Q}^2 as groups? I was pretty sure the answer was no, but it took me a little bit to come up with a proof. Of course, as one is supposed to, one sees that the argument that proves this really only has to do with the fact that \mathbb{Q} as a group, is the localization of some commutative unital ring and the fact that we’re discussing the localization as a module over the original ring. So, in the spirit of quicker, simpler post I record this fact here.

\text{ }

Continue reading

January 31, 2012 Posted by | Algebra, Group Theory | , , , , , , , | 2 Comments

Homomorphisms Between Finitely Generated Abelian Groups (Pt. I)


Point of Post: In this post we derive the necessary information to describe \text{Hom}_\mathbb{Z}(A,B) for any finitely generated abelian groups A,B according to their cyclic group decomposition.

\text{ }

Motivation

In this post, using our ability to split Hom across finite products we shall show how to effectively calculate the Hom group between any two finitely generated abelian groups. The only caveat to this is that one must first decompose the finitely generated abelian groups into their cyclic decomposition as is guaranteed by the structure theorem. Probably the most useful aspect of this post is that it will, if you keep your bookkeeping orderly, the number of homomorphisms between two finite abelian groups (presented as a product of cyclic groups) and the generators that will enable one to, in theory, produce every single homomorphism.

\text{ }

Continue reading

November 14, 2011 Posted by | Algebra, Group Theory, Module Theory, Ring Theory | , , , , , , , , , , , | 2 Comments

Homomorphism Group (Module) of Two Modules (Pt. I)


Point of Post: In this post we discuss how given two left R-modules one can construct the set \text{Hom}_R(M,N) of all R-morphisms M\to N and how this is naturally an abelian group, and how under the right conditions (commutativity of R) it is also a left R-module. Also, we define the notion of an R-algebra (a R-module which is also a ring, with multiplication compatible with scalar multiplication) and show how under the same assumptions the set \text{End}_R(M) is naturally a R-algebra.

\text{ }

Motivation

We know from the study of vector spaces that an extremely important object associated to every k-vector space V was the dual space V^\ast of all linear functionals V\to k.  Of course, after a little notation one is able to renotate the dual space as the set \text{Hom}(V,k) of all k-morphisms V\to k when k is considered as a one-dimensional vector space over itself. In fact, not long after this one realizes that given any other k-space W one can form the set \text{Hom}(V,W) of k-morphisms V\to W, which turns out to be a vector space of dimension \dim_k(V)\dim_k(W) when \dim_k(V),\dim_k(W)<\infty. We saw that the studying of these spaces were not only useful for outward looking facts about linear algebra (e.g. direct applications of them, using them for their own sake) but also gave us structural information about V as a vector space. Indeed, one can recall that \text{Hom}(V,F) had the ability to detect the finite-dimensionality of V since V\cong \text{Hom}(V,F) if and only if \dim_k(V)<\infty. This shall be a recurring fact that we shall be discussing extensively in the math-to-come, to be particular, we will see that a general R-module M can be studied quite extensively by watching how the “function” (functor) \text{Hom}_R(M,\bullet) as \bullet varies over other left R-modules. Consequently, we begin (just the very beginning) looking at \text{Hom}_R(M,N) for two left R-modules. We show that it’s always an abelian group, under certain conditions a left R-module, and when insist that M=N so that we get the set \text{End}_R(M) of endomorphisms it’s (under the same previously mentioned conditions) an R-algebra (module which is also a ring). We also then discuss (practically just define) the “function” (functor) we previously mentioned and show some of its most basic properties (i.e. that it is a functor).

\text{ }

Continue reading

November 1, 2011 Posted by | Algebra, Module Theory, Ring Theory | , , , , , , , , , , , , , | 6 Comments

Unit Group of a Finite Field is Cyclic


Point of Post: In this post we prove the basic fact that if k is a finite field then U(k) is cyclic.

\text{ }

Motivation

A very useful fact in both field and group theory is that if k is a finite field then the group of units U(k) is cyclic of order |k|-1. There are several ways one may go about doing this. In particular, there are two striking group theoretic facts about U(k) which one hope characterize it. Namely, from basic field theory we know for any polynomial p(x)\in k[x] one has that the number of roots of p(x) in k is bounded above by \deg(p(x)). In particular, using only the operation of the group we have that x^n=1 has at most solutions in k. The other interesting fact, which is easily implied by the bounded amount of roots x^n=1, is that U(k) has at most one cyclic subgroup of a given order (this is clearly implied by the previous property since if |g|=n and C_n is the cyclic subgroup generated by g then by Lagrange’s Theorem x^n=x^{|C_n|}=1 for each x\in C_n and so, in particular C_n constitutes n solutions of x^n=1 and so clearly there can’t be another distinct cyclic subgroup of order n). So, we prove that both of these properties characterize cyclic groups and so, in particular, U(k) is cyclic.

\text{ }

Continue reading

September 21, 2011 Posted by | Algebra, Group Theory | , , , , , , , , | 4 Comments

Sylow’s Theorems Revisited


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

\text{ }

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.

\text{ }

Continue reading

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

A Clever Proof of a Common Fact


Point of Post: In this post we give a new proof that if G is a finite group H is a subgroup G whose index is the smallest prime dividing |G| then that subgroup is normal.

\text{ }

Motivation

It is a commonly used theorem in finite group theory that if G is a finite group and H\leqslant G such that \left[G:H\right]=p is the smallest prime dividing |G| then H\unlhd G. We have already seen a proof of this fact by considering the homomorphism G\to S_p which is the induced map from G acting on G/H by left multiplication, and proving that \ker(G\to S_p)=H. We now give an even shorter (and the just mentioned proof is already short) proof of this fact using double cosets.

\text{ }

Continue reading

September 16, 2011 Posted by | Algebra, Group Theory | , , , , , , , , | Leave a comment

Actions by p-Groups (Pt. II)


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

Continue reading

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.

\text{ }

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.

\text{ }

Continue reading

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

A Classification of Integers n for Which the Only Groups of Order n are Cyclic (Pt. II)


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

Continue reading

September 13, 2011 Posted by | Uncategorized | , , , , , , , , | Leave a comment

A Classification of Integers n for Which the Only Groups of Order n are Cyclic (Pt. I)


Point of Post: In this post we conglomerate and extend a few exercises in Dummit and Foote’s Abstract Algebra which will prove that the only positive integers n for which the only group (up to isomorphism) of order n  is \mathbb{Z}_n are integers of the form n=p_1\cdots p_m are distinct primes with p_i\not\equiv 1\text{ mod }p_j for any i,j\in[m].

\text{ }

Motivation

This post will complete several lemmas/theorems which works towards proving not only that every group of order p_1\cdots p_m where p_i\not\equiv 1\text{ mod }p_j for any i,j\in[m] (greatly generalizing the statement that a group of pq for primes p<q with q\not\equiv 1\text{ mod }p is cyclic) but also that numbers of this form are the only numbers for which the converse is true (namely every group of order n is cyclic).

Continue reading

September 13, 2011 Posted by | Uncategorized | , , , , , , , , , , | 1 Comment