Let p be a prime and let .
Prove that if is a surjective group homomorphism and G is not the trivial group , then G is isomorphic to .
subgroup of with we have we have proved before that we must have for some now define by see that is a
well-defined homomorphism. it's also surjective. finally which is again obvious because: so we've proved that is a
surjective homomorphism and its kernel is thus by the first isomorphism theorem for groups we must have