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Proof on isomorphisms

  • Thread starter mariab89
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Homework Statement



Prove that for any isomorphism [tex]\phi[/tex] : G--> H |[tex]\phi[/tex](x)| = |x| for all x in G. is the result true if [tex]\phi[/tex] is only assumed to be a homomorphism?

Using the solution to the above proof or otherwise, show that any 2 isomorphic groups have the same number of elements of order n, for every positive integer n.

Homework Equations



The definition of a homomorphism states that given two arbitrary groups G and H. A function f : G---> H is called a homomorphism if f(ab) = f(a)f(b) for all a, b in G.

The Attempt at a Solution



What I started doing was that since we know that [tex]\phi[/tex] is an isomorphism, we know that it is bijective and a homomorphism. Therefore,
[tex]\phi[/tex](ab) = [tex]\phi[/tex](a)[tex]\phi[/tex](b) (because it is a homomorphism)

I'm not too sure what I can do from here...
 
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Answers and Replies

  • #2
Office_Shredder
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Do this in two parts

If the order of x is k, show that [tex] \phi(x)^k = e[/tex] e the identity. Then show that if the order of [itex] \phi(x) [/itex] is k, that xk is also the identity element
 

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