Compute the indicated quantities for the given homomorphism¢. (See Exercise 46.)

Ker(∅) and∅(20) for∅: Z → S₈ such that ∅(1) = (1, 4,2, 6)(2, 5, 7)

Data  from Exercise 46

Let a group G be generated by { a_i | i ∈ I}, where I is some indexing set and a_i ∈ G for all i ∈ I. Let ∅ : G → G' and µ : G → G' be two homomorphisms from G into a group G', such that ∅(a_i) = µ(a_i) for every i ∈ I. Prove that ∅ = µ. [Thus, for example, a homomorphism of a cyclic group is completely determined by its value on a generator of the group.]