## Sylow’s Theorems Revisited

**Point of Post: **In this post we give a more refined proof of Sylow’s Theorems.

*Motivation*

On this blog I have given a proof of Sylow’s theorems and an alternate proof of Sylow’s first theorem. A year or so later, with more time, and more finesse I’d like to give the simplest, most coherent proof of Sylow’s theorems.

*Sylow’s Theorems*

We assume all the notations such as being the set of Sylow -subgroups of and ,etc. are known to the reader:

**Theorem (Sylow): ***Let be a finite group with where , then:*

**Proof:**

We construct such a group inductively. Evidently if then we may take the trivial group. Suppose then that we have found a subgroup of with for . We know then that and so by a previous theorem we have that . Thus, by Cauchy’s Theorem there exists with . By the fourth isomorphism theorem we know that for some and so, in particular, we have that . As stated, from this we may construct subgroups of with .

Let act on by left multiplication. Since is a -group we have by the ‘fundamental theorem’ that . That said, since we have in particular that and so is non-empty. So, let then for all and so for all , or equivalently . Thus, . In particular, if we have that and so .

Let act on by conjugation. By we know that this action is transitive. In particular, we have that for any . That said, we note by definition that and so .

For the second part of the statement we let act on by conjugation. By the ‘fundamental theorem’ we know that . What we claim though is that . Indeed, it’s evident that , and so now assume that . We have then that and so , but we know that

and since and is the maximal power of dividing we may conclude that and so . Thus, . Noting then that completes the proof.

**References:**

1. Isaacs, I. Martin. *Finite Group Theory*. Providence, RI: American Mathematical Society, 2008. Print.

2. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

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