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-Special p groups | -Special p groups | ||
| − | + | -Pro p-groups | |
| − | + | -Powerful p-groups | |
Sylow Theorems -Application | Sylow Theorems -Application | ||
| Line 23: | Line 23: | ||
P-groups: | P-groups: | ||
| − | + | '''Definitions:''' | |
| + | |||
| + | *Let p be a prime p <math>\in</math> <math>\mathbb{Z}</math> such that <math>\mathbb{Z}</math>≥0. A p-group is a group of order p<sup>n</sup>. | ||
| + | *A subgroup of order p<sup>k</sup> for some k ≥ 1 is called a p-subgroup. | ||
| + | *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.<br> | ||
[[Category:MA453Fall2013Walther]] | [[Category:MA453Fall2013Walther]] | ||
Revision as of 14:00, 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
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 pk for some k ≥ 1 is called a p-subgroup.
- If |G| = pαm where p does not divide m, then a subgroup of order pα is called a Sylow p-subgroup of G.
