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

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

Definitons:

Definitions:

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