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Tuesday, November 3, 2009

Watched lecture 5 of Abstract Algebra E-222

Defines the equivalence relation on a set as a partition in disjoint subsets whose union is the set.
Properties of an equivalence relation:
- reflexive: a~a
- symmetric: a~b <=> b~a
- transitive: a~b and b~c => a~c.

A homomorphism f: G->H with kernel K which is a normal subgroup of G implies an equivalence relation on G where K is one of the equivalence classes. The other equivalence classes have the form aK = { ak; k in K, for some a in G}. aK is also called a left coset of K. ( Gross writes complete proof of this proposition on board. )
A bit of mathematical history about Lagrange ( born in Italy! ) who writes a letter to Euler at age 17 containing some very sophisticated mathematics. Euler immediately recognizes the genius of Lagrange and arranges further education for Lagrange who until that time learned his math through self-study.
(The famous) Theorem of Lagrange.
If G is a finite group and H is a subgroup of G then the order of H divides the order ( size ) of G.
More propositions are discussed.
- Groups of order p are simple.
- Groups of order p^2 are abelian.
- An is simple for n>=5.
- Any finite, non-abelian group has even order.

( Next lecture Peter. )

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