Written Assignment No. 2

as corrected in class on Oct. 10

due October 14, 2005

General Directions: Written assignments should be submitted typeset. What you submit must represent your own work.

Assigned Exercises

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.

1. Any group homomorphism ϕ : G → G′ where |G| is a prime must either be the trivial homomorphism or else be an injective (one-to-one) map.

2. For any groups G, G′, any subgroup H of G, any group homomorphism ϕ : G → G′, and any element a ∈ G one always has the equality of sets

{x ∈ G | ϕ(x) = ϕ(a)} = Ha .