Another homomorphism question

1. Nov 22, 2004

cmurphy

Let G be a finite group of even order with n elements. H is a subgroup of G, with n/2 elements.

I need to show that H is normal. I have set up the phi function to be: phi(x) = 1 if x is an element of H. phi(x) = -1 if x is not an element of H. Thus H is the kernal.

I am trying to show that H is a homomorphism. Then from that I know that H is normal.

I am breaking this up into cases. I have been successful showing that the homomorphism holds if x,y are both in H. It also holds if x is in H and y is in G.

However, I am having difficulty showing that if x and y are both in G, then their product xy must be in H. How do I go about showing that?

Colleen

2. Nov 23, 2004

matt grime

The thing you're trying to show is false, so I'd stop trying to show it.

Just consider the cosets G/H and let G act on them. Let the cosets be [e] and [x], show that the action of G on them by multiplication gives a homomorphism to C_2 and its kernel is H. Remember cosets are equal or disjoint.