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.
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”.
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
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.
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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.