Difference between revisions of "Walther453Fall13 Topic2" - Rhea

Revision as of 10:43, 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 \mathlarger{a, b} $\in$ $\mathlarger{G}$ , 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 51: / Line 51: @@
 *If |G| = p<sup><span class="texhtml">α</span></sup>m where p does not divide m, then a subgroup of order p<sup><span class="texhtml">α</span></sup> is called a Sylow p-subgroup of G.
+\usepackage{relsize}
 ==Regular p-groups==
 '''Definitons:'''