A Very Basic Introduction to the Sylow Theorems.

[Note: It’s been a while!  I’ve now completed most of my pre-research stuff for my degree, so now I can relax a bit and write up some topics.  This post will be relatively short just to “get back in the swing of things.”]

 

In Group Theory, the big question used to be, “Given such-and-such a group, how can we tell what it is?” 

 

 

The Sylow Theorems (proved by Ludwig Sylow, above) provide a really nice way to do this for finite groups using prime decomposition.  In most cases, the process is quite easy.  We’ll state the theorems here in a slightly shortened form, but you can read about them here.  Note that subgroup which is of order for some is unsurprisingly called a -subgroup. A -subgroup of maximal order in is called a Sylow -subgroup.

 

Theorem (Sylow).  Let be a group such that for .  Then,

1. There exists at least one subgroup of order . 
2. The Sylow -subgroups are conjugate to one-another; that is, if are Sylow -subgroups, then there is some such that .  Moreover, for all , we have that is a Sylow -subgroup.
3. The number of Sylow -subgroups of , denoted by , is of the form .  In other words, divides . 

 

This first part says that the group of Sylow -subgroups of is not empty if divides the order of .  Note that this is slightly abbreviated (the second part is actually more general, and the third part has a few extra parts) but this will give us enough to work with.

 

Problem: Given a group  for prime and , is ever simple (does it have any nontrivial normal subgroups)?  Can we say explicitly what is?

 

We use the third part of the Sylow theorems above.  We note that and , but so this immediately implies that (why?).  So we have one Sylow -subgroup; let’s call it .  Once we have this, we can use the second part of the Sylow theorem: since for each we have is a Sylow -subgroup, but we’ve shown that is the only one there is!  That means that ; this says is normal in .  We have, then, that isn’t simple.  Bummer.

On the other hand, we can actually say what this group is.  So let’s try that.  We know the Sylow -subgroup, but we don’t know anything about the Sylow -subgroups.  We know that and , but that’s about it.  There are two possibilities: either or . 

For the first case, by using the modular relation, if does not divide then this forces ; this gives us a unique normal Sylow -subgroup .  Note that since the orders of our normal subgroups multiply up to the order of the group, we have ; in other words, .

For the second case, .  We will have a total of subgroups of order and none of these are normal.   This part is a bit more involved (for example, see this post on it), but the punch line is that it will be the cyclic group .  

 

I’ll admit that the last part is a bit hand-wavy, but this should at least show you the relative power of the Sylow theorems.  They also come in handy when trying to show something either does or does not have a normal subgroup.  Recall that a simple group has no nontrivial normal subgroups.

 

Question.  Is there any simple group with ?  

 

I just picked this number randomly, but it works pretty well for this example.  We note that .  Let’s consider, for kicks, .  We know must divide and it must be the case that ; putting these two facts together, we get .  This immediately gives us a normal subgroup of order 11, which implies there are no simple groups of order 165.

 

Question.  Is there any simple group with ? 

 

Alas, alack, you may say that 777 is too big of a number to do, but you’d be dead wrong.  Of course, .  Use the same argument as above to show there are no simple groups of this order.

 

Question.  Is there any simple group with ? 

 

Note that so we need to do a little work, but not much.  Just for fun, let’s look at .  We must have that it is 1 modulo 7 and it must divide .  Hm.  A bit of thinking will give you that , which gives us the same conclusion as above.

 

Of course, there are examples where this doesn’t work nicely.  Think about the group of order 56, for example.  In order to work with these kinds of groups, one must do a bit more digging.  We will look into more of this later.

Tensor Products: A few explicit calculations.

Reader Beware.

I planned to do a post about tensor products (what they are, why we should care, what we do with them, etc.) but because I’m not comfortable with all of that quite yet, I’m going to assume you know what tensor products are, and do a few explicit calculations.  So, in short, if you don’t already know what tensor products are, don’t read this post.

Our notation will be as follows: is a field, is a commutative ring with , and will denote the tensor product of modules over a ring .  As usual, will denote the polynomials in with coefficients in .

(Note:  My thanks to Brooke, who pointed out that I kept writing "+" when I meant "."  I hope I’ve not made this error elsewhere, as tensors are "pretty different" from standard addition.)

Algebras, or When Can I Multiply Stuff in My Module?

Wordy Introduction, Motivation.

When you first start high school algebra, the big thing is FOIL-ing, right?  Factoring and factorizing quadratics.  When you get to calculus, the big things are derivatives and integrals.  Then when you get to college and start doing math, things get a little tougher.  We start learning about abstract structures, and these become increasingly specific and increasingly complex as we go along.

What the Hell is a Module?

This post is going to be a gentle introduction to what a module is.  It isn’t hard, but, for me, modules were sort of just “thrown in” with a whole bunch of defining properties and no motivation for why I should care about them.  I’m hoping to motivate them at least a little bit so that you feel more comfortable thinking and working with them!

Applying Lagrange!: Groups of Prime Orders.

Little post.  Because I love doing things that comments tell me to do, we’re going to use Lagrange to prove a neato theorem.  Now, normally, if I told you, “Hey, guy, I’ve got a group with elements.  What one is it?” you’d probably be unable to tell me!  Why?  Lots of different groups have the same order!  For example, if we’re talking about order 8, are we talkin’ ?  Are we talkin’ ?  Are we talkin’ ?  I just don’t know!

Lagrange’s Theorem: or how I learned to stop worrying and modulo out by a subgroup.

How could I have been so naive?  How could I have been so myopic?  How is it that I thought I could just wrap up group theory without mentioning Lagrange’s theorem?  How could I let this topic die out not with a bang but with a whimper?

Let us, for old time’s sake, state one more theorem for the group theory primer — and this one’s a biggie!  Remember how division is defined for rational numbers?  sort of means “split into little piles of size , and is how many piles there are.”  For example, if we have 12 batteries and put them into piles of 3 batteries each, how many piles do we have?  This doesn’t take a rocket scientist.

Group Theory Primer, part 5: the last of the isomorphism theorems.

Last time we talked about a whole lot of stuff.  We did homomorphisms, isomorphisms, and talked about the first ismorphism theorem.  What did this one state?  It states that for are groups and is a homomorphism, then we have that , or, in other words, the quotient of with the kernel of the map is equal to the image of the map.  This makes sense if you think about it: we’re kind of condensing everything that goes to 0 when we map it away from and we say that these elements ultimately don’t matter in the image — but, because of the nice properties of homomorphisms, a lot of other elements map onto each other, too.

Today, we’re going to discuss the final two isomorphism theorems (which don’t come up as often, but they’re nice) and conclude with one of the most used theorems in elementary abstract algebra: Cauchy’s Theorem.

Group Theory Primer, part 4: everything you wanted to know about homomorphisms but were afraid to ask.

Last time we went over some normal subgroups, how to direct product two groups, and how to quotient out by (normal) subgroups.  As we said before, though, groups (like vector spaces) are pretty boring by themselves.  Yes, studying groups by themselves can give us relations between elements and so on (like what kinds of elements in a particular group have the property such that when you square them they become the identity), but, like vector spaces, we can learn a lot about a group by what it can and can’t map into nicely.

Now, let’s think about this for a second.  What if I said something like the following: let’s take a group such that the elements are and .  Let’s say that

and those are all the possible interactions.  You could give any reasonable justification to this group, but it reduces to the fact that it is just a group: it’s just a set of elements and an operation.

Group Theory Primer, part 3: direct products, cosets, and quotients!

If we think about groups as if they were numbers, we’d want to add, subtract, multiply, and divide stuff.  Unfortunately, groups aren’t as simple as numbers, and we have more complex notions of what all of these things should correspond to.

Group Theory Primer, part 2: examples of a few groups.

Last time, we talked about what a group is.  This time, we’ll go over some specific groups.  In the next post, we’re going to go over some basic theorems about groups.