Difference between revisions of "Walther453Fall13 Topic2" - Rhea

Revision as of 14:20, 29 November 2013

Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu

Outline:

Origin -Creator -History of the Sylow Theorems/ p-groups P-Groups -Definition -Regular p-groups

               -Relationship to Abelian Groups

-Application -Frattini Subgroup

       -Special p groups
       -Pro p-groups
       -Powerful p-groups

Sylow Theorems -Application

                -Theorem 1

-Theorem 2 -Theorem 3

       -Importance of Lagrange Theory

I plan on deleting everything above this after we have completed the paper.  I planned on just using the outline as a guide. 

I've been using these websites: 

http://math.berkeley.edu/~sikimeti/SylowNotes.pdf

http://omega.albany.edu:8008/Symbols.html (this is Tex symbols)

The pdf emailed to you

http://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf

P-groups

Definitions:

• Let p be a prime p $ \in $ $ \mathbb{Z} $ such that $ \mathbb{Z} $≥0. A p-group is a group of order pⁿ.
• A subgroup of order p^k for some k ≥ 1 is called a p-subgroup.
• If |G| = p^αm where p does not divide m, then a subgroup of order p^α is called a Sylow p-subgroup of G.

Sylow's Theorems

Definitions:

Let G be a group of order p^αm, where p is a prime, m≥1, and p does not divide m.  Then:

1. Syl_p(G) 

Regular p-groups:

Definitions

• Suppose p is a prime number. A p-group G