Groups and kernels?

1. Jan 13, 2008

pivoxa15

1. The problem statement, all variables and given/known data
Expain why every normal subgroup is the kernel of some homomorphism.

3. The attempt at a solution
Every kernel is a normal subgroup but the reverse I can't show rigorously. It seems possible how to show?

2. Jan 13, 2008

morphism

What's the obvious map from G to G/N? What's its kernel?

3. Jan 13, 2008

HallsofIvy

Staff Emeritus
I would have thought that would be easy- it's the direction emphasised in Algebra texts! Of course, this says "some" homomorphism- you have to pick the homomorphism carefully.

If H is a normal subgroup G, then we can define the "quotient group", G/H. There is an obvious homomorphism from G to H. What is the kernel of that homomorphism?

Darn, I had to stop in the middle to take a telephone call and morphism got in in front of me!