General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.
Read these directions carefully: for each of the following statements either provide a proof that the statement is true or label the statement as false and provide justification.
Any group homomorphism phi : G --> G' where |G| is a prime must either be the trivial homomorphism or else be an injective (one-to-one) map.
For any groups G, G', any subgroup H of G, any group homomorphism phi : G --> G', and any element a \in G one always has the equality of sets
| {x \in G | phi(x) = phi(a)} = Ha . |