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.

1. 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 \(k\cdot Stab(s)\).

2. 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.

3. 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:

4. 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:

5. 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 \([G:H]\).

6. 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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