Thread: Proof question about homomorphism 1?

(a) Prove that for any isomorphism φ: G-->H, |φ(x)|=|x| for all x∈G. Is the result still true if φ is only assumed to be a homomorphism?
(b) Using (a) or otherwise show that any two isomorphic groups have the same number of elements of order n, for every positive integer n.

Originally Posted by koukou8617

(a) Prove that for any isomorphism φ: G-->H, |φ(x)|=|x| for all x∈G. Is the result still true if φ is only assumed to be a homomorphism?
(b) Using (a) or otherwise show that any two isomorphic groups have the same number of elements of order n, for every positive integer n.

For (a), what have you done about the first part? For the second part, you need a counter-example. Clearly, the kernel of your mapping must be trivial (why?) and so your mapping must be 1-1. If it is an injective homomorphism but NOT an isomorphism, what then can you say?

For (b) you need to remember that your isomorphism is also a bijection, and so the two groups contain the same number of elements. Each element in the image has the same order as in the pre-image, and so they must tally.