Abstract
If one wishes to construct a homomorphism of a group, an obvious approach is to begin with a homomorphism of a subgroup and attempt to extend it in some way. Any such extended homomorphism is determined by its action on elements of the group which, together with the subgroup, generate the whole group. The values assigned to such generators must then satisfy various functional equations. By a governing list for the extension problem, we shall mean a list of functional equations whose solutions describe all possible homomorphisms extending a given subgroup homomorphism.
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