Question

Only need answer from (IV) to (VI)

Only need answer from (IV) to (VI)

Math 3140 page 1 of 7 1. (30) Let R be the group of real numbers under addition, and let U = {e® : 0 E R} be the group of all complex numbers on the unit circle under multiplication. Let o: R U be the map given by = e is a homomorphism of groups. (i) Prove that (i) Find the kernel of . (Don't just write down the definition. You need to describe an explicit subset of R.) (iii Find a real number r for which p(r) = i. [Hint: i =e*i/2.] Math 3140 page 2 of 7 (iv) Find the image (or "range") of o. (v) Let G and H be groups and let : G H be a homomorphism of groups. State the Fundamental Homomorphism Theorem (First Isomorphism Theorem) as it applies to this situation (vi) Prove that the group Z (consisting of the integers under addition) is a normal subgroup of R (vii) Prove that the quotient group (or "factor group") R/Z is isomorphic to U.

Answer 1

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