(a) Let G be a group and a E G. If kEZ- {0} such that af...

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(a) Let G be a group and a E G. If kEZ- {0} such that af = e, prove that: (1) The order o(a) of a is finite. [4] (ii) The order of a divides k. [6] (b) Let G = (a) be an infinite cyclic group. Consider a map f : Z → G defined by f(k) = ak for all ke Z. (i) Prove that f is a group homomorphism. [4] (ii) Show that f is an isomorphism (i.e. every infinite cyclic group is isomorphic to Z). [6] (a) Let G be a group and a E G. If kEZ- {0} such that af = e, prove that: (1) The order o(a) of a is finite. [4] (ii) The order of a divides k. [6] (b) Let G = (a) be an infinite cyclic group. Consider a map f : Z → G defined by f(k) = ak for all ke Z. (i) Prove that f is a group homomorphism. [4] (ii) Show that f is an isomorphism (i.e. every infinite cyclic group is isomorphic to Z). [6]