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