Rank of a Group

In group theory, the rank of a group G is the smallest number of elements needed to generate the entire group, serving as a rough analogue of the dimension of a vector space. For a finitely generated group this rank is always a non-negative whole number. The trivial group has rank zero, and every nontrivial cyclic group has rank one; the free abelian group on n generators has rank n, and a free group has rank equal to the size of its defining basis. Finite non-abelian simple groups, such as the alternating groups on five or more letters, all have rank two. A related notion, subgroup rank, is the largest rank found among all of a group's subgroups, and no subgroup can exceed the rank of the whole group. Determining the rank of a group is decidable for finite groups, abelian groups and nilpotent groups, but the general rank problem remains unsolved for several other important classes of finitely presented groups. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/