Compute the indicated quantities for the given homomorphism. (See Exercise 46.) Ker() and (25) for :

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

Ker(∅) and ∅(25) for ∅ : Z → Z7 such that ∅(1) = 4

Data  from Exercise 46

Let a group G be generated by { ai | i ∈ I}, where I is some indexing set and ai ∈ G for all i ∈ I. Let ∅ : G → G' and µ : G → G' be two homomorphisms from G into a group G', such that ∅(ai) = µ(ai) 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.]

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