Question

Problem 6. The set (Z19 − {0}, ·19) is a group with the indicated operation; see the attached table. a.) Show that H = {1, 7, 8, 11, 12, 18} is a subgroup. b.) List all the right cosets of H. c.) Show that if Hy = Hx then xy−1 ∈ H. [Make sure to give a reason for each step.] d.) Show that φ : H → Hx defined by φ(h) = hx is one-to-one and onto. [Use the fact that H is a group and the property of inverses to do this. Part (c) may also come in handy.] e.) Show that the operation ⊗ defined by (Ha) ⊗ (Hb) ≡ Hab is well-defined.

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