In Exercises determine whether the given map ϕ is a homomorphism. [Hint: The straightforward way to proceed is to check whether ϕ (ab) = ϕ(a) ϕ(b) for all a and b in the domain of ϕ. However, if we should happen to notice that ϕ-1 [{e'}] is not a subgroup whose left and right cosets coincide, or that ϕ does not satisfy the properties given in Exercise 44 or 45 for finite groups, then we can say at once that ϕ is not a homomorphism.
Let ϕi : G1 → G1 × G2 × … × Gi × … × Gr be given by ϕi(gi) = (e1, e2,…, gi,…,er), where gi ∈ Gi and ej is the identity element of Gj. This is an injection map. Compare with Example 13.8.
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