Watched lecture 3 of Abstract Algebra E-222

( 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. )