text: In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
  In other words, if  is a subset of a group , then , the subgroup generated by , is the smallest subgroup of  containing every element of , which is equal to the intersection over all subgroups containing the elements of ; equivalently,  is the subgroup of all elements of  that can be expressed as the finite product of elements in  and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.)
  If , then we say that  generates , and the elements in  are called generators or group generators. If  is the empty set, then  is the trivial group , since we consider the empty product to be the identity.
  When there is only a single element  in ,  is usually written as . In this case,  is the cyclic subgroup of the powers of , a cyclic group, and we say this group is generated by . Equivalent to saying an element  generates a group is saying that  equals the entire group . For finite groups, it is also equivalent to saying that  has order .