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Outline:  
 
Outline:  
  
Origin 
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Origin   
  
 
-History of the Sylow Theorems/ p-groups  
 
-History of the Sylow Theorems/ p-groups  
  
P-Groups -Definition
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P-Groups -Definition  
  
-Regular p-groups
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-Regular p-groups  
  
 
-Relationship to Abelian Groups  
 
-Relationship to Abelian Groups  
  
-Application
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-Application  
  
-Frattini Subgroup
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-Frattini Subgroup  
  
<br>
+
<br>  
  
Sylow Theorems -Application<br>
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Sylow Theorems -Application<br>  
  
 
=== I plan on deleting everything above this after we have completed the paper. &nbsp;I planned on just using the outline as a guide.&nbsp;<br>  ===
 
=== I plan on deleting everything above this after we have completed the paper. &nbsp;I planned on just using the outline as a guide.&nbsp;<br>  ===
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<br> <br>  
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= Email the group to see if anyone else is currently making changes before you begin making changes yourself!!!<br> <br> =
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 +
 
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== P-groups  ==
 
== P-groups  ==
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<br>  
 
<br>  
  
All proofs of these Propositions can be found [http://www.ms.unimelb.edu.au/~ram/Notes/pgroupsSylowtheoremsContent.html here]  
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All content from this section and proofs of these Propositions can be found [http://www.ms.unimelb.edu.au/~ram/Notes/pgroupsSylowtheoremsContent.html here]  
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== Regular p-groups  ==
 
== Regular p-groups  ==
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#n<sub>p</sub>(G) is congruent to 1 mod p.&nbsp;
 
#n<sub>p</sub>(G) is congruent to 1 mod p.&nbsp;
  
All Proofs of these Theorems can be found [http://math.berkeley.edu/~sikimeti/SylowNotes.pdf here]  
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All content from this section and&nbsp;proofs of these Theorems can be found [http://math.berkeley.edu/~sikimeti/SylowNotes.pdf here]  
  
 
<br>  
 
<br>  

Revision as of 12:30, 30 November 2013

Mark Rosinski, markrosi@purdue.edu Joseph Lam, lam5@purdue.edu Beichen Xiao, xiaob@purdue.edu

Outline:

Origin 

-History of the Sylow Theorems/ p-groups

P-Groups -Definition

-Regular p-groups

-Relationship to Abelian Groups

-Application

-Frattini Subgroup


Sylow Theorems -Application

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 regular p-group

http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf this one is alm[[|]]ost about everything.



Email the group to see if anyone else is currently making changes before you begin making changes yourself!!!

P-groups

Definitions:

  • Let p be a prime p be an integer greater or equal to 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.


Propositions:

If G is a p-group then G contains an element of order p.

  1. If G is a p-group then Z(G)cannot be equal to {1}
  2. Let p be a prime and let G be a group of order p2. Then G is abelian.
  3. If G is a p-group of order pa, then there exists a chain, {1} is contained in N1 contained in N2 contained in...contained in Na-1 contained in Gof normal subgroups of G, such that |Ni|=pi.


All content from this section and proofs of these Propositions can be found here



Regular p-groups

Definitons:

  • For every $ a, b \in G $ there exists $ c \in [<a,b>,<a,b>] $ such that apbp = (a'b)pcp
  • For every $ a, b \in G $ there exist $ c_1 , c_2 , . . . , c_k \in [<a,b>,<a,b>] $ such that $ a^p b^p = (ab)^p c^p _1 c^p _2 . . . c^p _k $
  • For evert $ a, b \in G $ and every natural number n there exist Failed to parse (syntax error): c_1 , c_2 , . . . , c_k \in {,a,b>,<a,b>]
such that $ a^q b^q = (ab)^q c^q _1 c^q _2 . . . c^q _k $ where q = pn



Sylow's Theorems

Notation:

Sylp(G) = the set of Sylow p-subgroups of G

np(G)= the # of Sylow p-subgroups of G =|Sylp(G)|

Theorems:

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

  1. Sylp(G) cannot be the empty set. 
  2. All Sylow p-subgroups are conjugate in G. To expand, if P1 and P2 are both Sylow p-subgroups, then there is some g in G such that P1=gP1g-1.                           In particular, np(G)=(G:NG(P)).
  3. Any p-subgroup of G is contained in a Sylow p-subgroup
  4. np(G) is congruent to 1 mod p. 

All content from this section and proofs of these Theorems can be found here




Extra Information

For students looking for extensive history on p-groups, Sylow's Theorems and finite simple groups in general you can find this information here

Alumni Liaison

To all math majors: "Mathematics is a wonderfully rich subject."

Dr. Paul Garrett