Normal Subgroups: Why Every Kernel is a Homomorphism

[pivoxa15]

Homework Statement

Expain why every normal subgroup is the kernel of some homomorphism.

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?

 

[morphism]

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

 

[HallsofIvy]

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!