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DON U] 10. 11. 12. 13. . Find an isomorphism from the group of integers under addition to the group of even integers under addition. . Find Aut(Z). . Let R+ be the group of positive real numbers under multiplication. Show that the mapping qb(x) : V)? is an automorphism of R+. . Show that U(8) is not isomorphic to U(1{}). . Show that U(8) is isomorphic to U02). . Prove that isomorphism is an equivalence relation. That is, for any groups G, H, and K G== G; G\"=HimpliesH== G GmHandeKimpliesGmK. . Prove that S 4 is not isomorphic to D12. . Show that the mapping a > log10 a is an isomorphism from RJr under multiplication to R under addition. . In the notation of Theorem 6.1, prove that I; is the identity and that (Tg '1 2 T3_,. Given that (,'b is a isomorphism from a group G under addition to a group G under addition, convert property 2 of Theorem 6.2 to addi- tive notation. Let G be a group under multiplicatioth be a group under addition and (the an isomorphism from G to G. If (,'b(a) = E and (15(1)) = 5, nd an expression for qb(a3b '2) in terms of E and 3. Let G be a group. Prove that the mapping 04g) : g'1 for all g in G is an automorphism if and only if G is Abelian. If g and h are elements from a group, prove that (g), 2 g