Homomorphisms, finite groups, and primes

[kathrynag]

Homework Statement

1. Let G and H be finite groups and let a: G → H be a group homomorphism. Show
that if |G| is a prime, then a is either one-to-one or the trivial homomorphism.
2. Let G and H be finite groups and let a : G → H be a group homomorphism. Show
that if |H| is a prime, then a is either onto or the trivial homomorphism.

Homework Equations

The Attempt at a Solution

1. We know a(b)a(c)=a(bc) since it is a homomorphism
order is prime.
need to show a(x1)=a(x2) implies x1=x2. I'm confused on how the oder being prime plays into this.

 

[Outlined]

if |G| is prime then it is a cyclic group generated by one single element. Hope this helps.

hint: write f for your homomorphism and not a.

 

[jbunniii]

You don't need to mess around with elements. Hint:

The kernel of a homomorphism from G to H is a _______ of G
The image of a homomorphism from G to H is a _______ of H

 

[annoymage]

hmm, may i ask what trivial homomorphism means?

 

[Outlined]

The homomorphism which maps everything to 0.

 

[kathrynag]

Since it's a cyclic group generated by one element it must be one to one since there is only element.

|H| is prime. H is a cyclic group generated by one element. Must have x such that f(x)=y