http://akbar.marlboro.edu/~mahoney/courses/Spr00/rubik.html said:
Group Generators
Given a group G (and in this section, we’ll almost always use Rubik’s group R as our group),
then if S ⊂ G is any subset of the group, then the subgroup H generated by S is the smallest
subgroup of G that contains all the elements of S.
Surely such a subgroup exists. If any collection of different subgroups contain all the members
of S, then their intersection (which is also guaranteed to be a subgroup) also contains all the elements of S and is contained in all of them. The group G is a subgroup of itself, so there is at
least one group in that intersection.
Intuitively, the subgroup generated by S is the collection of all the elements you can get to
by repeatedly multiplying members of S or their inverses together. In the case of R, it’s the set
of all positions you can arrive at, starting with the jumbled cube and applying only that subset of
moves in any order.
For example, the subgroup of R generated by { F } contains four members. If you’re only
allowed to turn the right face, you can only get to four different cube configurations. The subgroup
generated by { FF } is even smaller: two positions. In other words, if you’re only allowed
to make half-turns of one face, there are only two possible positions you can achieve starting
from solved.
Obviously, the group generated by { F, B, R, L, U, D } is the entire group R—just a small
number of generators can generate a huge group. We are interested, of course, in subgroups that
fall between the extremes mentioned in this and the previous paragraph.