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*A subgroup of order p<sup>k</sup> for some k ≥ 1 is called a p-subgroup. | *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. | *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. | ||
| + | |||
| + | ==Regular p-groups== | ||
| + | '''Definitons:''' | ||
| + | *Suppose p is a prime number. A p-group G (i.e., a group where the order of every element is a power of p) is termed a regular p-group if it satisfies the following equivalent conditions: | ||
| + | 1. For every a,b \in G, there exists c \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc^p. | ||
| + | 2. For every a,b \in G, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc_1^pc_2^p \dots c_k^p. | ||
| + | 3. For every a,b \in G and every natural number n, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^qb^q = (ab)^qc_1^qc_2^q \dots c_k^q where q = p^n. | ||
| + | |||
<br> | <br> | ||
Revision as of 15:35, 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
Contents
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
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.
Regular p-groups
Definitons:
- Suppose p is a prime number. A p-group G (i.e., a group where the order of every element is a power of p) is termed a regular p-group if it satisfies the following equivalent conditions:
1. For every a,b \in G, there exists c \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc^p. 2. For every a,b \in G, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^pb^p = (ab)^pc_1^pc_2^p \dots c_k^p. 3. For every a,b \in G and every natural number n, there exist c_1,c_2, \dots, c_k \in [\langle a,b\rangle, \langle a,b \rangle] such that a^qb^q = (ab)^qc_1^qc_2^q \dots c_k^q where q = p^n.
Sylow's Theorems
Definitions:
Let G be a group of order pαm, where p is a prime, m≥1, and p does not divide m. Then:
- Sylp(G)
