Difference between revisions of "Walther453Fall13 Topic2" - Rhea

Revision as of 11:21, 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

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 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.

Propositions:

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

If G is a p-group then Z(G)cannot be equal to {1}
Let p be a prime and let G be a group of order p². Then G is abelian.
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 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 $a p b p = (a' b) p c p$
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  $q = p n$

Sylow's Theorems

Notation:

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

n_p(G)= the # of Sylow p-subgroups of G =|Syl_p(G)|

Theorems:

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

Syl_p(G) cannot be the empty set.
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, n_p(G)=(G:N_G(P)).
Any p-subgroup of G is contained in a Sylow p-subgroup
n_p(G) is congruent to 1 mod p.

All 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

@@ Line 10: / Line 10: @@
          -Special p groups
-   -Pro p-groups
+ -Pro p-groups
-   -Powerful p-groups
+ -Powerful p-groups
 Sylow Theorems -Application
@@ Line 33: / Line 33: @@
 and also the pdf emailed to you
-http://groupprops.subwiki.org/wiki/Regular_p-group  regular p-group
+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 almost about everything.
+http://people.maths.ox.ac.uk/craven/docs/lectures/pgroups.pdf this one is alm[[|]]ost about everything.
+<br>
 ----
+<br> <br>
-<br>
 == P-groups  ==
@@ Line 47: / Line 47: @@
 '''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>.
+*Let p be a prime p be an integer greater or equal to 0.&nbsp;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>
+'''Propositions:'''
+If G is a p-group then G contains an element of order p.
+#If G is a p-group then Z(G)cannot be equal to {1}
+#Let p be a prime and let G be a group of order p<sup>2</sup>. Then G is abelian.
+#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.
-<math>\usepackage{relsize}</math>
-==Regular p-groups==
-'''Definitons:'''
-* For every <math>a, b \in G</math> there exists <math>c \in [<a,b>,<a,b>]</math> such that <math>a^p b^p = (ab)^p c^p</math>
-* For every <math>a, b \in G</math> there exist <math>c_1 , c_2 , . . . , c_k \in [<a,b>,<a,b>]</math> such that <math>a^p b^p = (ab)^p c^p _1 c^p _2 . . . c^p _k</math>
-*For evert <math>a, b \in G</math> and every natural number <math>n</math> there exist <math>c_1 , c_2 , . . . , c_k \in [<a,b>,<a,b>]</math> such that <math>a^q b^q = (ab)^q c^q _1 c^q _2 . . . c^q _k</math> where <math>q = p^n</math>
 <br>
+All proofs of these Propositions can be found [http://www.ms.unimelb.edu.au/~ram/Notes/pgroupsSylowtheoremsContent.html here]
+== Regular p-groups  ==
+'''Definitons:'''
+*For every <math>a, b \in G</math> there exists <math>c \in [<a,b>,<a,b>]</math> such that <span class="texhtml">''a''<sup>''p''</sup>''b''<sup>''p''</sup> = (''a''''b'''''<b>)<sup>''p''</sup>''c''<sup>''p''</sup></b></span>
+*For every <math>a, b \in G</math> there exist <math>c_1 , c_2 , . . . , c_k \in [<a,b>,<a,b>]</math> such that <math>a^p b^p = (ab)^p c^p _1 c^p _2 . . . c^p _k</math>
+*For evert <math>a, b \in G</math> and every natural number <span class="texhtml">''n''</span> there exist '''Failed to parse (syntax error): c_1 , c_2 , . . . , c_k \in {,a,b&gt;,&lt;a,b&gt;]'''
+ such that <math>a^q b^q = (ab)^q c^q _1 c^q _2 . . . c^q _k</math> where <span class="texhtml">''q'' = ''p''<sup>''n''</sup></span>
+<br> <br>
 == Sylow's Theorems  ==
-'''Definitions:'''
+Notation:
+Syl<sub>p</sub>(G) = the set of Sylow p-subgroups of G
+n<sub>p</sub>(G)= the # of Sylow p-subgroups of G =|Syl<sub>p</sub>(G)|
+'''Theorems:'''
 Let G be a group of order&nbsp;p<sup>α</sup>m, where p is a prime, m≥1, and p does not divide m. &nbsp;Then:<sub></sub>
-#Syl<sub>p</sub>(G)&nbsp;<sub></sub><math>\neq</math>'''Failed to parse (lexing error): \0'''
+#Syl<sub>p</sub>(G) cannot be the empty set.&nbsp;
-All Sylow p-subgroups are conjugate in G. To expand, if P<sub>1</sub> and P<sub>2</sub> are both Sylow p-subgroups, then there is some g<sup></sup>
+#All Sylow p-subgroups are conjugate in G. To expand, if P<sub>1</sub> and P<sub>2</sub> are both Sylow p-subgroups, then there is some g in G such that P<sub>1</sub>=gP<sub>1</sub>g<sup>-1</sup>.<sup></sup>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;In particular, n<sub>p</sub>(G)=(G:N<sub>G</sub>(P)).
+#Any p-subgroup of G is contained in a Sylow p-subgroup
+#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]
 <br>
+<br>
+<br>
+== 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 [http://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00909-0/S0273-0979-01-00909-0.pdf here]
 [[Category:MA453Fall2013Walther]]

Difference between revisions of "Walther453Fall13 Topic2" - Rhea

Revision as of 11:21, 30 November 2013

Contents

I plan on deleting everything above this after we have completed the paper. I planned on just using the outline as a guide.

P-groups

Regular p-groups

Sylow's Theorems

Extra Information

Alumni Liaison