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 p^k for some k ≥ 1 is called a p-subgroup.
• If |G| = p^αm 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.

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 above and proofs of these Propositions can be found here

Further Information on p-groups:

• Lie Algebras
  • A lie ring is a set R with two binary operations - addition and the Lie bracket - such that
    • (R,+) is an abelian group;
    • The bracket operation distributes over addition;
    • [x,x] = 0 for all x in R;
    • [[x,y],z]+[[y,z],x]+[[z,x],y]=0 for all x,y,z in R.
  • If F is a field, and R is an F-vector space with a[x,y]=[ax,y] then R is a Lie algebra.
  • To every finite p-group one can associate a Lie ring L(G), and if G/G' is abelian then L(G) is actually a lie algebra over the finite field GF(p).
    • Proposition: Let φ be an automorphism of the finite p-group G. Then φ induces an automorphism on L(G), and if φ has order prime to p, then the induced automorphism has the same order.

• Number of Groups
  • Let g(n) denote the number of groups of order n.
    • i) g(p)=1 for p a prime.
    • ii) if p<q, then g(pq)=1 if q is not congruent to 1 mod p, and g(pq)=2 otherwise.
    • iii) g(p2)=2.
    • iv) g(p3)=5.

From this we can see that the number of groups of order n depends more on the prime structure then on its size.

Look at this graph to help explain this notion:

Regular p-groups

Definitons:

• For every $ a, b \in G $ there exists $ c \in [<a,b>,<a,b>] $ such that a^pb^p = (a'b)^pc^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 = pn

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:

1. Syl_p(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, n_p(G)=(G:N_G(P)).
3. Any p-subgroup of G is contained in a Sylow p-subgroup
4. n_p(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