Let g be a group and let phi s3 rarr g be a non-trivial

Question

1. Let R^x = R - {0} and set G = R^x x R. Define * on G by the following:

(a, b)*(c, d) = (ac, bc + d)

Show that G is a group under *. Is it abelian?

2. Let G be a group and let φ: S₃ → G be a non-trivial homomorphism. Suppose φ is not injective. Find the kernel of φ. If φ is surjective, what is G?

3. Assume that G is a group with a subgroup H such that |H| = 6, [G : H] > 4 , and |G| < 50 . What are the possibilities for |G|?

4. Define new addition ⊕ and multiplication ? on Z by

a ⊕ b= a + b - 1, and a ? b = a + b - ab

where operations on the right hand side of the equal signs are ordinary addition, subtraction and multiplication. Show that Z with these new operations, ⊕ and ?, is an integral domain.