( Off-topic: Since I am ' in between jobs ', which happens if you are a freelance IT professional and there in an economic crisis, I should be studying new Oracle features or something like that. Instead I watched another Algebra lecture, well the day is still young. )

Watched lecture 3 of Abstract Algebra E-222. ( A lecture by Peter, Gross's assistant, if he isn't a professor yet, he will be soon, I suppose. )

Review of lectures 1 and 2.

- Groups. And examples of groups GL(n,R), Sn, Z+.

- Subgroups. Cyclic subgroups.

- ( Hom(Rn, Rn) has the structure of a vectorspace. )

- All subgroups of Zn are of the form bZ. ( Emphasis on importance of proof of this proposition.)

- ( Studying the subgroup structure of a group is in general very difficult. )

- Example of a cyclic subgroup of GL(2,R). The group generated by {{1 1}, {0,1}} is {{1 n}, {0,1}} n in Z.

Example of an isomorphism.

G1 = {i, -1, -i, 1}

G2 = {(1,2,3,4}, (1,3),(2,4), (1,4,3,2), ()}

G1 and G2 are isomorphic by i -> (1,2,3,4)

( Permutations are here in cyclic notation which are not introduced in the course yet. )

Example of an isomorphism.

G1 = {R,+}

G2 = {R\{0},*}

G1 and G2 are isomorphic by f: G1->G2; x |-> e^x

Proof:

f(x+y)=e^(x+y)=e^x * e^y = f(x)*f(y).

Klein4 group.

V={() , (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)} as a subgroup of S4.

V={ {{1 0}, {0,1}}, {{-1 0}, {0,1}}, {{1 0}, {0,-1}}, {{-1 0}, {0,-1}} as a subgroup of GL(2,R).

Definitions.

-Automorphism.

-Homomorphism.

-Image ( of a homomorphism)

Next lecture Gross on images of homomorphisms ( and more ).

( Thank you, Peter. )

1-2017 More on the randomness of randomness.

2 months ago

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