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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.2. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 1 4 5 6 7 8 10 11 12. 13. 14 15 17. 18. 214 2] 2 8 10 12. 14 18. 1 3 5 7 9 11

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OR SU Ha {17 7,8, 11, 12, 18} Here 1 is the identity element also 7.8= 18 7.11=1, 7.12=8 7.18= 12 11.18=8 8.11 = 12 8.12=1, 8

e for cosets (Ha) ® H b) = Hab Now for a,bt (249-{0}, in) abe (2,9 -403, ig) Then Ha Hb and Hab all, are cosets of H Again we

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