ABSTRACT
A group with a prime number of elements can have no proper, nontrivial subgroups. Componentwise structure on product sets, as was studied in the preceding sections, is one rich source of new semigroups, monoids, and groups. Another is found from subsets that are closed under the given structure. In order to study the structure, the key properties of the sets of functions are abstracted and formulated in general terms. Starting with given examples of semigroups, monoids, or groups, there are methods to obtain new semigroups, monoids, or groups from the given ones. A semigroup S of functions may also provide the underlying set for an abstract semigroup structure with a multiplication which is different from the composition of functions.
