Group Theory – Direct Products

One of the ways of producing (potentially) new groups from old is called the direct product.Let me show you the direct product of two groups by example, first.

Let me invoke the abstract algebra package and set it to “groups”.

C2 x C2

I want two copies of C2, the cyclic group of order 2…

… and I am going to ask for their “direct product”.

What did we get?

First of all, we see that the elements of the direct product are just ordered pairs – the first component from H, the second component from K.


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Happenings – 2012 Feb 26

I decided to play hooky from school yesterday, so here’s the homework that was due yesterday.

By around 10 AM, I had a fairly simple happenings post laid out in my head.

3 books came in during the week. Lorenz’ “The Essence of Chaos” begins fairly slowly, but I think it will be a worthwhile read. Mumford’s “The Red Book of Varieties and Schemes” looks excellent… it has a list of prerequisites which I do not have, but which tell me what I need to get out of my current study of rings… he says in the preface that “The weakness of these notes is what had originally driven me to undertake the bigger project: there is no real theorem in them!” Instead, “The hope was to make the basic objects of algebraic geometry as familiar to the reader as the basic objects of differential geometry and topology….”

Sounds like a valuable resource.

Bloomfield’s “Fourier Analysis of Time Series” is fascinating. It’s fairly small, 250 pages… it may only have 3 significant examples… but I’ve already persuaded Mathematica to work out one of them – something other than the straightforward computation of the discrete Fourier transform of the data, followed by the periodogram. Instead, I did a nonlinear model which fitted frequencies directly. I’m sorry it’s taken me so long to get around to that.
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Group Theory: normal subgroups

I want to show you a simple concept. It has major consequences, but it starts out as a far simpler thing than it usually looks like.

I want to talk about normal subgroups. Clearly, if all subgroups were normal, we wouldn’t need to distinguish normal from non-normal… so I will have to show you a non-normal subgroup. That won’t be a problem.

Before I go into the details, let me give you the answer. Even if you know this material, the answer may surprise you.

A subgroup K of a group G is a normal subgroup of G if and only if K is the kernel of a homomorphism defined on G.

I know… I know: besides “normal”, I’ve used three other terms that I haven’t defined: subgroup, kernel, and homomorphism. If you’ve never seen this material before, you’re lost – but I can fix that.

Again, before I go into the details, let me give you the names of other things I will be talking about: fibers, left cosets and right cosets, and factor groups; I will briefly mention conjugacy classes.

My starting point will be to define a (group) homomorphism. It is just a function between groups which honors the group multiplications. (They do not need to be, and generally are not, the same operation in the 2 groups.)
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Happenings – 2012 Feb 18

Let’s see.

It’s been a relatively slow week, mathematically. You can see that a group theory post went out last Monday… I had said last Saturday morning that it existed only in my head. Clearly I succeeded in taking it from stage II through stage V… but I didn’t have time for much else. I’m a little disappointed in that, but rather pleased that I finished what I did.

My alter ego the kid looked at control theory… and my alter ego the graduate student is still looking at rings. It was my undergraduate who put out the group theory post. He will be trying to write another group theory post for this Monday.

Although the 2nd and 3rd group theory posts have not matched the success of the 1st one, business is still booming on Mondays and Tuesdays: total hits for the last 3 Mondays have been 530, 437, and 367; total hits for the last 3 Tuesdays has been 327, 331, and 332. Considering that the previous high was 309 in March of 2010, I’m delighted.

I’ve also got 3 books on order.

The kid had also looked at algebraic geometry recently, so I ordered Mumford’s “The Red Book of Varieties and Schemes…” republished with additional material.

Something – catastrophe theory? Korner’s “the pleasures of counting” or “naïve decision theory” – reminded me of something by Edward Lorenz … the Lorenz equations of a simple chaotic system.

And, since I’ve been thinking about Fourier analysis of time series, I ordered a well reviewed book of that title by Bloomfield. It turns out there was a recent 2nd edition.

And with that, it’s time to start writing a group theory post for Monday. I want to talk about normal subgroups. My 1st exposure to them was confusing as hell… but, to completely change the metaphor, I know how to cut the Gordian knot. They can be introduced very simply.

(If you know that a group is called simple if it has no proper normal subgroups, you may have cringed at the previous sentence. I swear that no pun was intended.)

Group Theory: Symmetric, Alternating, and Quaternion groups

Let’s talk about “symmetric groups” and, at the very end of the post, “alternating groups” and the “quaternion group”.

First, let’s get the abstract algebra package running… and tell it that we’re doing groups. (The alternative is “rings”, which have two operations instead of one.) You can find this free package here.

Let’s see what happens if I ask for the symmetric group “3” (and I’ve asked for its set of elements and called it p):

The help system tells me that the elements of this group are all of the permutations of 3 objects, with composition as the group product.
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Happenings – 2012 Feb 11

(This post is also in the “quaternions” category. If that’s what you’re looking for, just get the rest of the post and page down to the section heading.)

Let me begin by saying that the 2nd group theory post was not quite as popular as the 1st one… the blog “only” got 436 hits Monday, and the 2nd group theory post only got 102 hits itself. On Tuesday, as it had the week before, the blog as a whole got more than 300 hits.

I assure you I’m not disappointed in these kinds of numbers.

I hope to put out another post on group theory – about symmetric groups, permutations and cycles, and the quaternion group. On the one hand, the material is not complicated….

On the other hand, that post is at stage II. I’m not sure I’ve ever talked before about a specific post being at stage II. That means that I haven’t even done the mathematics for it! All I have is an outline in my head. We’ll see what happens.

As for the week gone by….
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Group Theory: Dihedral Groups

Having looked at cyclic groups, let’s look at dihedral groups. They can be created geometrically by starting with a cyclic group Cn… think of it as rotations of a regular n-gon through multiples of 360°/n… and then imagine that you can also spin the n-gon out of the plane about some axis.

Huh?

Let’s look back at the cyclic group C3. Take an equilateral triangle. Take our group operation r to be a rotation thru 120° (= 360°/3). Then r^2 is a rotation thru 240°, and r^3 is a rotation thru 360° – that is, r^3 = 1, the identity element.

Here a pictures of what r and r^2 do to an equilateral triangle:


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