In using the three isomorphism theorems, it is often necessary to know the actual correspondence given by the isomorphism and not just the fact that the groups are isomorphic. 

Let ∅: Z₁₈ →Z₁₂ be the homomorphism where ∅(1) = 10.

a. Find the kernel K of ∅. 

b. List the cosets in Z₁₈/K, showing the elements in each coset. 

c. Find the group [Z₁₈]. 

d. Give the correspondence Z₁₈/K and ∅[Z₁₈] given by the map μ described in Theorem 34.2.

Data from Theorem 34.2

Let ∅ : G → G' be a homomorphism with kernel K, and let _YK : G → G/K be the canonical homomorphism. There is a unique isomorphism µ: G/K → ∅[G] such that ∅(x) = µ,(γK(x)) for each x ∈ G. The lemma that follows will be of great aid in our proof and intuitive understanding of the other two isomorphism theorems.