11. Let H = {2k | ke Z}. Show that H ≤ Q*. 12. For N = 0, 1,2,... let NZ = {Nk | ke Z}. Prove that NZ ≤ Z and then show

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11 Let H 2k Ke Z Show That H Q 12 For N 0 1 2 Let Nz Nk Ke Z Prove That Nz Z And Then Show 1 (18.33 KiB) Viewed 13 times

11. Let H = {2k | ke Z}. Show that H ≤ Q*. 12. For N = 0, 1,2,... let NZ = {Nk | ke Z}. Prove that NZ ≤ Z and then show that these are in fact the only subgroups of Z. 13. Let G be a group and H, K≤ G. Prove that HK and HUK are subgroups of G. 14. Let G be a group and show that Z(G) = {x € G | xg = gr for all g € G} I is a subgroup of G. We call Z(G) the center of G.