# The Sylow Theorems

The Sylow theorems seek to understand the structure of groups. Knowing only the order of a group, what can one expect about the existence and the orders of the subgroups, their relationship to each other and their numbers?

I have a copy of the second edition of John B. Fraleigh’s “A First Course in Abstract Algebra” on hand, in which he emphasized over and over again:

“*Never underestimate a theorem which counts something.*”

Curiously Fraleigh chose to not include proofs for the Sylow Theorems in his book and the chapter was starred to indicate that it was not required for a one-semester course. Wikipedia has proofs for the theorems but is rather hard to follow. In fact these very important theorems are neither tedious nor hard to follow and there is a very nice technique that can be repeatedly used to prove them. I will mostly follow Nathan Carter’s excellent “Visual Group Theory”, which taught me to comprehend these theorems in a very intuitive and accessible way.

## Normalizer

Lagrange Theorem tells us that the order of a subgroup *H* divides that of the group *G*. For Abelian groups we can extend the concept of division to the groups themselves: the cosets of *H* form a quotient group *G/H*, where the group operation is defined by multiplying elements from two cosets of *H* together using the standard operations of *G* to obtain another coset of *H*. For non-Abelian groups we can only form the quotient group this way if the subgroup *H* is *normal*, that is any left coset of *H* is also its right coset. In a sense we relax the requirement of commuting from the element level to \(g\cdot H=H\cdot g\) instead. And indeed for a normal subgroup *H* we can define coset multiplication like the Abelian case:\[g_{1}H\cdot g_{2}H = (g_{1}\cdot g_{2})H\]

Not all subgroups are normal. For an arbitrary subgroup *H*, we define its *normalizer* to be the union of all the left cosets of *H* that are also its right cosets: \(N_{G}(H)\equiv \{g\in G\mid gH=Hg\}\).

It is easy to verify that \(N_{G}(H)\) is a subgroup of G. We can build the quotient group \(N_{G}(H)/H\) under coset multiplication as before. The order of the quotient group is noted as \([N_{G}(H):H]=\vert N_{G}(H)\vert/\vert H\vert\).

## Cauchy’s Theorem

Given a set *S* and a group *G* that acts on *S*, we say

the

*orbit*of an element*s*in the set is the subset that the group elements of*G*take it to: \(Orb(s) = \{x \in S \mid \exists g \in G, x=g \circ s\}\);*s*is*stable*if it is not moved by any group element, that is the size of its orbit is 1;the

*stabilizer*of an element*s*in the set is the subgroup of*G*whose elements do not change*s*: \(Stab(s) = \{g \in G \mid s=g \circ s\}\)

It is easy to check that: every element in *Orb(s)* shares the same orbit; having identical orbits is an equivalence relationship therefore orbits form a partition of the set *S*; and *Stab(s)* is indeed a subgroup.

**Theorem**(*Orbit-Stabilizer Theorem*) The size of an element’s orbit times the size of its stabilizer is the size of the group: \[\vert Orb(s)\vert \cdot \vert Stab(s)\vert = \vert G\vert\]*Proof*. In the case of \(\vert Orb(s)\vert=1\) we obviously have \(\vert Stab(s)\vert=\vert G\vert\). In general we can find a one-to-one mapping between*Orb(s)*and the left cosets of*Stab(s)*in*G*. Given any coset \(k\cdot Stab(s)\) every element in the coset takes*s*to the same element \(k\circ s\) in*Orb(s)*. Conversely, for any element*a*in*Orb(s)*we have a group element*k*that takes*s*to it. If*h*also takes*s*to*a*then \(h^{-1}\cdot k\) is in*Stab(s)*and therefore*h*is in the coset \(k\cdot Stab(s)\).**Theorem**If a group*G*of prime order*p*acts on a set*S*, then the order of*S*and the number of stable elements in*S*are congruent mod*p*.*Proof*. As*p*is prime, the size of any orbit can only be 1 or*p*. The number of non-stable elements, i.e. those in orbits of size*p*, is a multiple of*p*.**Theorem**(*Cauchy’s Theorem*) If*p*is a prime number that divides the order of the group*G*, then*G*has an element*g*of order*p*, and therefore a cyclic subgroup \(\langle g\rangle\) of order*p*.*Proof*. There exists one inverse element for the product of any tuple of*p–1*group elements. Therefore the set*S*of tuples of the form \(\left(a_{1}, a_{2},..., a_{p}\right)\) that satisfies \(a_{1}\cdot a_{2}\cdot...\cdot a_{p}=e\) has order \(\vert G\vert^{p-1}\) . If we rotate any tuple in*S*we obtain another tuple in*S*, so rotations form the cyclic group \(C_{p}\) that acts on*S*. Let*N*be the number of stable elements of*S*that are invariant under rotations, then by our last theorem we have:\[N\cong_{p}\vert S\vert\cong_{p}0\]We know \(N\gt 0\) since the tuple \((e, e, ..., e)\) is a stable element of*S*, so*N*is a nonzero multiple of*p*. There exists another stable element of*S*and let its form be \((g, g, ..., g)\), then \(g^{p}=e\) and*g*is the element we are looking for.

In proving Cauchy’s Theorem we used a powerful technique that we shall keep returning to: we find a group that acts on a set and we then seek to characterize the order of the set based on the actions of the group.

## First Sylow Theorem

A *p-group* (*p-subgroup*) is a group (subgroup) whose order \(p^{m}\) is the power of some prime number *p*. We can easily expand theorem (2) to:

**Theorem**If a*p-group G*acts on a set*S*, then the order of*S*and the number of stable elements in*S*are congruent mod*p*.

Furthermore we have:

**Theorem**If*H*is a*p-subgroup*of*G*, then \([N_{G}(H):H]\cong_{p}[G:H]\).*Proof*. We consider the action of*H*on the set*S*of the left cosets of*H*. The stable elements of*S*are precisely those left cosets of*H*who are also right cosets, the union of which is the normalizer \(N_{G}(H)\). The number of the stable elements under*H*is therefore the number of its cosets in \(N_{G}(H)\) which is \([N_{G}(H):H]\), while the order of*S*is the number of cosets of*H*in*G*which is \([G:H]\).**Theorem**(*First Sylow Theorem*) If*G*is a group and \(p^{n}\) is the highest power of a prime number*p*dividing \(\vert G\vert\), then there are subgroups of*G*of every order \(1, p, p^{2}, ..., p^{n}\). Also, every*p-subgroup*with fewer than \(p^{n}\) elements is inside a larger*p-subgroup*.*Proof*. Subgroup of order 1 trivially exists:*{e}*. By Cauchy’s theorem subgroup of order*p*also exists. We shall prove that for every subgroup*H*of size \(p^{i}, i\lt n\) we can construct another subgroup of*G*that contains*H*and that is*p*times as large. We note that \([G:H]=\vert G\vert/\vert H\vert\cong_{p}0\), therefore \([N_{G}(H):H]\cong_{p}0\). So*p*divides the order of the quotient group \(N_{G}(H)/H\). By Cauchy’s theorem there exists a subgroup of order*p*in the quotient group, whose elements are cosets of*H*. The union of these cosets is a subgroup of*G*that contains*H*with order \(p\vert H\vert=p^{i+1}\).

## Conjugacy and the Class Equation

## Second Sylow Theorem

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