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Midterm 2 – Math 363, instructor: Pawel Gladki Friday, August 10th, 2007.

Time: 60 minutes 1. Prove that the set

H = {a ∈ R∗: a3 is rational}

is a subgroup of R∗ (nonzero reals with multiplication).

2. Determine all left and right cosets of the subgroup H = {id, O120, O240} in the group D(3).

3. Determine orders of all the elements of the group U (Z12). Is U (Z12) cyclic?

4. Show that {I, −I} C GL(n, R), where I denotes the identity matrix.

5. Check if the following function is an endomorphism of the group C∗ (nonzero complex numbers with multiplication):

φ(z) = 5z.

1

Cytaty

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