Isomorphic Groups and Order of Elements

[mariab89]

Homework Statement

Prove that for any isomorphism $$\phi$$ : G--> H |$$\phi$$(x)| = |x| for all x in G. is the result true if $$\phi$$ 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 $$\phi$$ is an isomorphism, we know that it is bijective and a homomorphism. Therefore,
$$\phi$$(ab) = $$\phi$$(a)$$\phi$$(b) (because it is a homomorphism)

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

 

[Office_Shredder]

Do this in two parts

If the order of x is k, show that $$\phi(x)^k = e$$ e the identity. Then show that if the order of $\phi(x)$ is k, that x^k is also the identity element