How to think about Sylow’s Theorem

If you haven’t come across Sylow’s theorem yet, you’re in for a treat!

Should you’ve come across it, you might remember Sylow’s theorem as a confusing amalgamation of 3 theorems with technical proofs. I will try to present these theorems in a natural and easy-to-remember way. Note: Throughout this post, groups will be finite.

The first major theorem one encounters in abstract algebra is Lagrange’s theorem.

Lagranges Theorem copy
A consequence of Lagrange’s theorem is: In a finite group the order of every element divides the order of the group.1

So given the existence of a subgroup or element of a group, we know that its order must divide |G|, the order of the group.

But what about the converse? Given a divisor d of |G|, can we show the existence of a subgroup or element of order d in G?

The answer is no. For example: A4, the alternating group of degree 4, has order 12 but no subgroup of order 6.

Yet, mathematicians did not take “no” for an answer and tried to find a partial converse. The first such partial converse is Cauchy’s theorem.
Cauchys Theorem
Given the existence of an element g of order p, we deduce the existence of a subgroup of order p, namely the cyclic group generated by g.

However, cyclic groups of order p aren’t really that interesting. They behave too nicely! Thankfully, the Norwegian Ludwig Sylow stated and proved the following generalization of Cauchy’s theorem:
Sylows Theorem I
The existence of a group of order pn implies the existence of (normal) subgroups of orders pr for all r≥0.
Using the notation of the theorem, subgroups of order pn (i.e. maximal p-(sub)groups) are called Sylow p-subgroups, and P is such a Sylow p-subgroup.

Knowing about the existence of an object isn’t enough, as we are curious about its properties. Sylow’s second theorem states a crucial such property.

Sylows Theorem II copy
Explicitly, if P and Q are Sylow p-subgroups there exists an element g in G such that gPg-1 = Q.

Now if we consider the normalizer of a Sylow p-subgroup P, i.e. NG(P) = { g in G | gPg-1 = P}, then since it is a subgroup of G, we can think about its index |G:NG(P)|. As all Sylow p-subgroups are conjugate,  |G:NG(P)|= # Sylow p-subgroups, which is often denoted by np. Since P is a subgroup of NG(P), we deduce:

np · |NG(P):P| = |G:NG(P)||NG(P):P| = |G:P|=  pn m ÷ pnm, i.e. np | m.

Combining this with one more result, we get:

Sylows Theorem 3 copy

If this seems like too much information, just remember the following diagram which sums up the key ideas:

  1. Cauchy and Sylow’s Theorems are partial converses to Lagrange’s Theorem. (Indicated by the dotted arrows.)
  2. Sylow’s Theorem generalizes Cauchy’s Theorem. (Shown by red > orange.)

Summary Sylow copy

Save

Save

Save

1 Given an element g of a finite group G, let H be the cyclic subgroup generated by g, noting that |g|=|H|. Now apply Lagrange’s theorem to G and H.

Save

Save

Leave a Reply

Your email address will not be published. Required fields are marked *