# Math Help - order

1. ## order

If f: G --> H is an isomorphism, prove that |f(x)| = |x| for all x in G. Deduce that any two isomorphic groups have the same number of elements of order n for each positive integer n. Is it true "if f is a homomorphism, then |f(x)| = |x|?" If true, prove it. If false, give an example.

2. Originally Posted by dori1123
If f: G --> H is an isomorphism, prove that |f(x)| = |x| for all x in G. Deduce that any two isomorphic groups have the same number of elements of order n for each positive integer n. Is it true "if f is a homomorphism, then |f(x)| = |x|?" If true, prove it. If false, give an example.
Hint: Prove that if $f: G_1 \to G_2$ is an isomorphism then $|x|$ divides $|f(x)|$. Therefore if $f: G_1 \to G_2$ is an isomorphism then $f: G_1 \to G_2$ and $f^{-1} : G_2 \to G_1$ are homomorphisms therefore $|x|$ divides $|f(x)|$ and $|f(x)|$ divides $|x|$, so $|x|=|f(x)|$.