# 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.

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: *A _{4}*, 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.

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:

The existence of a group of order *p ^{n}* implies the existence of (normal) subgroups of orders

*p*for all

^{r}*r≥0*.

Using the notation of the theorem, subgroups of order

*p*(i.e. maximal

^{n}*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.

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.* N _{G}(P) =* {

*g*in

*G*|

*gPg*}, then since it is a subgroup of

^{-1}= P*G*, we can think about its index |

*G*:

*N*|. As all Sylow

_{G}(P)*p*-subgroups are conjugate,

**|**, which is often denoted by

*G*:*N*|= # Sylow_{G}(P)*p*-subgroups*n*. Since

_{p}*P*is a subgroup of

*N*, we deduce:

_{G}(P)*n _{p} · *|

*N*:

_{G}(P)*P*| = |

*G*:

*N*||

_{G}(P)*N*:

_{G}(P)*P*| = |

*G*:

*P*|=

*p*÷

^{n}m*p*=

^{n}*m*, i.e.

**.**

*n*|_{p}*m*Combining this with one more result, we get:

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

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

^{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*.