## A Very Basic Introduction to the Sylow Theorems.

### May 18, 2012

[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 is a group, how can we tell which group 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 $p^{\beta}$ for some $\beta$ is unsurprisingly called a $p$-subgroup. A $p$-subgroup of maximal order in $G$ is called a Sylow $p$-subgroup.

Theorem (Sylow).  Let $G$ be a group such that $|G| = p^{\alpha}m$ for $p\not| m$.  Then,

1. There exists at least one subgroup of order $p^{\alpha}$.
2. The Sylow $p$-subgroups are conjugate to one-another; that is, if $P,Q$ are Sylow $p$-subgroups, then there is some $g\in G$ such that $gPg^{-1} = Q$.  Moreover, for all $g\in G$, we have that $gPg^{-1}$ is a Sylow $p$-subgroup.
3. The number of Sylow $p$-subgroups of $G$, denoted by $n_{p}$, is of the form $n_{p} \equiv 1\mbox{ mod } p$.  In other words, $n_{p}$ divides $m$.

This first part says that the group of Sylow $p$-subgroups of $G$ is not empty if $p$ divides the order of $G$.  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  $|G| = pq$ for $p,q$ prime and $p < q$, is $G$ ever simple (does it have any nontrivial normal subgroups)?  Can we say explicitly what $G$ is?

We use the third part of the Sylow theorems above.  We note that $n_{q} | p$ and $n_{q} \equiv 1\mbox{ mod } q$, but $p < q$ so this immediately implies that $n_{q} = 1$ (why?).  So we have one Sylow $q$-subgroup; let’s call it $Q$.  Once we have this, we can use the second part of the Sylow theorem: since for each $g\in G$ we have $gQg^{-1}$ is a Sylow $q$-subgroup, but we’ve shown that $Q$ is the only one there is!  That means that $gQg^{-1} = Q$; this says $Q$ is normal in $G$.  We have, then, that $G$ 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 $Q$-subgroup, but we don’t know anything about the Sylow $P$-subgroups.  We know that $n_{p} \equiv 1\mbox{ mod }p$ and $n_{p} | q$, but that’s about it.  There are two possibilities: either $n_{p} = 1$ or $n_{p} = q$.

For the first case, by using the modular relation, if $p$ does not divide $q-1$ then this forces $n_{p} = 1$; this gives us a unique normal Sylow $p$-subgroup $P$.  Note that since the orders of our normal subgroups multiply up to the order of the group, we have $PQ \cong P\times Q \cong G$; in other words, $G \cong {\mathbb Z}_{p}\times {\mathbb Z}_{q}$.

For the second case, $n_{p} = q$.  We will have a total of $q$ subgroups of order $p$ 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 ${\mathbb Z}_{pq}$.

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 $|G| = 165$?

I just picked this number randomly, but it works pretty well for this example.  We note that $|G| = 165 = 3\cdot 5\cdot 11$.  Let’s consider, for kicks, $n_{11}$.  We know $n_{11}$ must divide $3\cdot 5 = 15$ and it must be the case that $n_{11} \equiv 1\mbox{ mod } 11$; putting these two facts together, we get $n_{11} = 1$.  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 $|G| = 777$?

Alas, alack, you may say that 777 is too big of a number to do, but you’d be dead wrong.  Of course, $777 = 3\cdot 7\cdot 37$.  Use the same argument as above to show there are no simple groups of this order.

Question.  Is there any simple group with $|G| = 28$?

Note that $28 = 2^{2}\cdot 7$ so we need to do a little work, but not much.  Just for fun, let’s look at $n_{7}$.  We must have that it is 1 modulo 7 and it must divide $2^{2} = 4$.  Hm.  A bit of thinking will give you that $n_{7} = 1$, 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.