Walther453Fall13 Topic2 - Rhea

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)

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

and also the pdf emailed to you

http://groupprops.subwiki.org/wiki/Regular_p-group regular p-group

http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf this one is almost about everything.

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.

$ \usepackage{relsize} $

Regular p-groups

Definitons:

• For every $ a, b \in G, there exists c \in $ $ \in $$ G $, there exists $ \mathlarger{c} $$ /in $$ \mathlarger{{[<a,b>,<a,b>]} $

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) $ \neq $Failed to parse (lexing error): \0

All Sylow p-subgroups are conjugate in G. To expand, if P₁ and P₂ are both Sylow p-subgroups, then there is some g