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Sunday, December 20, 2009

Lecture 21 of Abstract Algebra E-222

Again the Sylow theorems and a summary of the proofs.
Then finite groups of certain orders were classified.
Order p, p prime. Cyclic groups of order p.
Order p * q, p > 2, p < q. Cyclic of order p*q and if p / (q-1): Cq : Cp.
Order 2p, cyclic of order 2p and the Dihedral groups Dp.
Order p^2*q. Started with order 12:
- C12
- C6 X C2
- A4
- D6
- C4 : C3.
I can add that
- A4 = ( C2 X C2 ) : C3.

Gross sofar never talked about GAP, Mathematica, Magma or Maple. Group theory can get very complicated without the aid of a mathematics package. The groups were classified solely on the basis of the Sylow theorems in this lecture. The semi-direct product has not been lectured yet. For those new I would say these lectures were real hard. Unnecessarily hard in my opinion. The next lecture is about the Symmetry group by a famous number theory guy ( forgot his name ).

Anyway, I discovered an interesting fact about S6 today, the symmetry group on 6 letters. It is the only symmetric group whose automorphism group is not eqaul to the group itself. - Now, why would that be? Seems like some deep fact to me. - This is were mathematics gets a grip on you. You must know why.

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