1. The problem statement, all variables and given/known data I want to find the kernel of PHI: Z-> Z (mod 24) X Z (mod 81) I am beginning to think that the kernel of this is actually just the set containing the identity element, or the trivial subgroup of Z mod 24. I am thinking this because none of the subgroups of Z mod 24 are equal to the identity of Z mod 81. Am I thinking about this correctly??