## Review of Group Theory: Alternate Proof of the Sylow Theorems

**Point of post: **In this post we give some alternate proofs of Sylow’s theorems which use a little more machinery then the “classic” ones.

*Motivation*

As remarked the Sylow Theorems are probably the most fundamentally important set of theorems in finite group theory. Consequently, any new proof verifying their validity is welcomed. In this post we give alternate proofs (in some sense) of the Sylow theorems than the “classic” proofs we have already given. They are less constructive and use more machinery. That said, they are very elegant and “cute”.

*Sylow’s Theorems*

We begin with a simple enough number theoretic lemma:

**Lemma: ***Let be prime and . Then, for all one has*

**Proof: **We note that since is prime it’s trivial that

for . Thus,

(recalling that two polynomials are equivalent if and only if their coefficients are equivalent). It clearly follows using the same argument that

and so proceeding by induction one finds that

for every . In particular

and so

comparing the coefficients of gives the desired result.

With this we can prove what is the equivalent statement of the first Sylow theorem in our last post.

**Theorem: ***Let be a group and a prime such that . Then, has a Sylow -subgroup.*

**Proof: **For notational convenience let where . We then let

Note that is evidently a -space under the -action of left multiplication, namely for each and define

Note by the Orbit Decomposition Theorem that

and thus we must have, by our lemma, that for some . It follows then that

from where it follows that . Note though that if then and thus . It follows then that and thus . We may then conclude that from where the conclusion follows.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 200

3.Grillet, Pierre A. *Abstract Algebra*. New York: Springer, 2007. Print.

[…] Point of post: This is a continuation of this post. […]

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