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.
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...