Order of image conjugacy class divides conjugacy class?

  • #1
I have a surjective group homomorphism ψ:G → G', and I've shown that if an element x in G has conjugacy class C and its image ψ(x) has conjugacy class C', then ψ restricts to a surjective map from C to C'.

Now I'd like to show that the order of C' divides C. I know from the class equation that |C| divides |G|, that |C'| divides |G'|, and that due to surjectivity, |G'| divides |G|, so |C| divides |G|. I can't see how to piece these facts together to show that |C'| divides |C| though, or how to use what I proved in the first part, so there's definitely something I'm missing.

Any help is appreciated!
 

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