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