Question 3 In each case below, verify whether the described algebraic structure is a group. Check the axioms and do not forget to verify whether the given object is indeed an algebraic structure! In each case, additionally check whether it is commutative – either provide a coun- terexample or argue that it it is indeed commuative (verify this axiom even if it is not a group, but only if it is an algebraic structure). Justify your answer and show all work. (i) (R²,⋆), where (x,y) ⋆ (a,b) = (x+a, y −b). (Here R² = {(z,w) : z, w ≤ R}.) (ii) (R\ {0},0), where a ob=a².b². (iii) The complex numbers of complex norm 1 under the usual complex multiplication. (iv) The set of all 2 × 2 upper-triangular matrices having determinant 1, under the usual matrix multiplication. (v) The set Q[x] of all polynomials in x with coefficients from Q and under the usual addition of polynomials.

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