5. G = {a + b2: a eZ and be Z) is a group under addition in the reals. Define o: G G for all a+ b./2 eG, as (a+ b /2) = a- b./2. Prove that o is an automorphism.

Q: on G by a * b = ba. Prove that G is isomorphic

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Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

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Q: 1. Let A be an abelian group, and let G = A × A (Cartesian product), which also is a group. Let 6:…

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Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

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