## Review of Group Theory: Sylow’s Theorems

**Point of post: **In this post we discuss the concept of Sylow’s Theorems.

**Note: **A considerably slicker, and more coherent version of these proofs is given in this post of mine.

*Motivation*

The majority of the interesting theorems in basic finite group theory can be proven using Sylow’s Theorem (or their ilk) in one way or another. That said, not much motivation is needed.

*Cauchy’s Lemma for Abelian Groups*

We begin with a lemma of Cauchy which will help us greatly along the way. In essence Cauchy’s Lemma says that if is a finite abelian group and a prime such that then has an element of order . More precisely:

*Remark: *In the following proof keep in mind the convention that for an order of an element and are used interchangeably.

**Theorem: ***Let be a finite abelian group and a prime such that . Then, contains an element of order . *

**Proof: **It suffices to prove this for a fixed but arbitrary prime . We induct on abelian groups of order where . Clearly the result is true for since implies that is cyclic and thus even non-identity element of has order . Assume the result is true for and assume that . If is cyclic we’re done since every cyclic group has precisely one subgroup of each order which divides the ambient group (see here for a proof). So, assume that is not cyclic and let be such that . If we’re done since would have an element of order so assume not. Then, and since is an abelian group whose order is of the form we may apply the induction hypothesis to product some such that . Note though that if is the canonical projection then using the fact that for any homomorphism we may conclude that

But, this implies by previous theorem that

Thus, is an element of of order . The induction is complete.

**Sylow’s Theorems**

With Cauchy’s Lemma we are now able to prove the Sylow theorems. First though we need a little terminology.

Let be a finite group and be such that where is prime and . Then, is called a *Sylow -subgroup. *In general a group is called a *-group *if for every , for some ( is dependent upon ). A subgroup of a group which is a -group is called a *-subgroup *of .

**Theorem( First Sylow’s Theorem): ***Let be a finite group and let be a prime. If then has a subgroup of order where .*

**Proof: ** We do this by induction on . Clearly the result is true for . Assume then that every group of order has the property that if then any group of order has a subgroup of order and let be a group of order . If then by Cauchy’s Lemma we have that there exists some such that . Note then that and . Thus, by the induction hypothesis there exists some subgroup such that . Recall though that one of the consequences of the Fourth Isomorphism Theorem is for some . Note though that

and thus the conclusion follows.

Suppose now that then evidently by the class equation

and thus by assumption we must have that there exist some for which . Note then though that we must have that otherwise we’d see from

that contradictory to our assumption. Thus and since the induction hypothesis implies that and consequently has a subgroup of order . The induction is complete.

**Corollary(Cauchy’s Theorem): ***Let be a finite group and a prime such that . Then, contains an element of order .*

**Proof: **By the First Sylow’s Theorem we know that has a subgroup of order and since this subgroup must be cyclic (see here for proof) we know that for some and for and thus contains at least elements of order . Since the conclusion follows.

**Corollary: ***Let be a finite group. Then is a -group if and only if for some .*

**Proof: **Evidently if then clearly is a -group. Conversely, let be a -group and suppose that is a prime other than such that . Then, by Cauchy’s theorem there exists some such that . But, clearly cannot then be a power of contradicting that is a -group. It follows if then for some . Thus, for some .

**References:**

1. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge: Cambridge UP, 1994. Print.

2. Lang, Serge. *Undergraduate Algebra*. 3rd. ed. Springer, 2010. Print.

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

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

Pingback by Review of Group Theory: Sylow’s Theorems (Pt. II) « Abstract Nonsense | January 8, 2011 |

[…] 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 […]

Pingback by Review of Group Theory: Alternate Proof of the Sylow Theorems « Abstract Nonsense | January 11, 2011 |

[…] Every -group is […]

Pingback by Review of Group Theory: Solvable Groups « Abstract Nonsense | March 11, 2011 |

[…] show that if is a simple group of order where and are primes then it is abelian. Indeed, by Sylow’s first theorem we know that must have a subgroup of order . If has only one of these such subgroups, then by […]

Pingback by Representation Theory: Burnside’s Theorem « Abstract Nonsense | March 11, 2011 |

[…] it happens, finagling around with the conditions on the number of Sylow -subgroups of some group that one finds that for every prime that […]

Pingback by Relation Between Sylow’s Theorems and Direct Product « Abstract Nonsense | April 19, 2011 |

[…] (Pure Group Theory): Let be a finite group and a Sylow -subgroup, then if and only if […]

Pingback by Groups of Order pq (pt. I) « Abstract Nonsense | April 19, 2011 |

[…] Let denote the number of Sylow -subgroups of . By Sylow’s theorems we must have that and so but since we must conclude that . Thus, […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) ( January 2003)) « Abstract Nonsense | May 1, 2011 |

[…] we note that if the maximum power of that divides is that (since is an odd prime) and so by Sylow’s theorems that must contain a Sylow -subgroup of order . But, by part a) we have that is a normal Sylow […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2003) « Abstract Nonsense | May 1, 2011 |

[…] By Sylow’s Theorems we know that has a subgroup of order , and by Sylow’s theorems has a subgroup of order , […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2004) « Abstract Nonsense | May 6, 2011 |

[…] trivially from the structure theorem. That said, there is a more simpleminded proof. Namely, by Cauchy’s Theorem there exists with . That said, it’s trivial fact of group theory that if one has a […]

Pingback by A Classification of Integers n for Which the Only Groups of Order n are Cyclic (Pt. I) « Abstract Nonsense | September 13, 2011 |

[…] the most significant use of actions by -groups is given by a clever proof of Cauchy’s Theorem given by McKay. […]

Pingback by Actions by p-Groups (Pt. II) « Abstract Nonsense | September 15, 2011 |