Difference between revisions of "Walther453Fall13 Topic2" - Rhea

Revision as of 09:49, 30 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

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^$α$m where p does not divide m, then a subgroup of order p^$α$ is called a Sylow p-subgroup of G.

$\usepackage{relsize}$

Definitons:

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

Definitions:

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

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

@@ Line 1: / Line 1: @@
- \documentclass{article}
- \usepackage{relsize}
- \begin{document}
- \[
-    x+y=z \leftrightarrow \mathlarger{\mathlarger{\mathlarger{(x+y=z)}}}
- \]
- \end{document}
 Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu
@@ Line 64: / Line 55: @@
 ==Regular p-groups==
 '''Definitons:'''
-* For every \mathlarger{a, b} <math>\in</math><math>\mathlarger{G}</math>, there exists <math>\mathlarger{c}</math><math>/in</math><math>\mathlarger{{[<a,b>,<a,b>]}</math>
+* For every <math>a, b</math> <math>\in</math><math>G</math>, there exists <math>\mathlarger{c}</math><math>/in</math><math>\mathlarger{{[<a,b>,<a,b>]}</math>