let Φ be an isomorphism from G to Ḡ prove that |a| = |Φ(a)| for all a in G.?

Answer 1

here are some hints

you know phi(ab) = phi(a) phi(b)

so phi(a^2) =phi(a) phi(a) = phi(a) ^2.

generalize this.

also phi(e) = e