## Friday, December 28, 2012

### Aargh! The Ordeal of Too Much Math Homework!

The poor college student above is overwhelmed! The math is too difficult! Who would have believed that  a teacher- to -be has to become immersed in Abstract Algebra, Geometry and Linear Algebra? It's too much!  Now, "Phoebe" needs your help! The 3rd year Education major is stuck on an Abstract Algebra problem.

Let's peek in on her preliminary reading and see what the problem is:

Let (G, o) and (H, o) be groups. Then a homomorphism of (G, o) into (H, o) is a map of the sets G and H which has the following property:  f(x o y) = f(x) o f(y)

Example:

(G, o) = (R1 +)

(H, o) = (R*, ·)

Take f = the exponential function, so f(x) = exp (x), f(y) = exp(y)

Then: f(x + y) = exp(x + y) = exp(x) exp(y) = f(x) f(y)

Or:

H = R* = {x Î R: x  not equal 0}

And: exp R -> R* so exp(x + y) = exp(x) exp(y)

Def.: Isomorphism: An isomorphism of G onto H [(G, o), (H, o)] is a bijective homomorphism.

Example: H = P = {x Î R: x > 0}         (P, x)

Let G = (0, 1, 2, 3) for the operation (o) which is addition in Z4

Let H = (2, 4, 6, 8) for the operation (o) which is multiplication in Z10

Problem: Prepare the respective tables for the isomorphism and give specific examples in terms of the function φ, i.e. show specific mappings.  (Where: φ(x) φ(y) = φ(xy) for example)

Can you help Phoebe out with her homework?

Hint: Check out an earlier blog on isomorphisms: http://www.brane-space.blogspot.com/2011/02/isomorphism-between-galois-groups-and.html

All proposed solutions welcome as comments!