Homomorphisms, finite groups, and primes

  • Thread starter kathrynag
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  • #1
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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.
 

Answers and Replies

  • #2
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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.
 
  • #3
jbunniii
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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
 
  • #4
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hmm, may i ask what trivial homomorphism means?
 
  • #5
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The homomorphism which maps everything to 0.
 
  • #6
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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
 

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