Rank of a group

In the mathematical subject of group theory, the rank of a group "G", denoted rank("G"), can refer to the smallest cardinality of a generating set for "G", that is
If "G" is a finitely generated group, then the rank of "G" is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for "p"-groups, the rank of the group "P" is the dimension of the vector space "P"/Φ("P"), where Φ("P") is the Frattini subgroup.
The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group "G" is the maximum of the ranks of its subgroups:
Sometimes the subgroup rank is restricted to abelian subgroups.
The rank problem.
There is an algorithmic problem studied in group theory, known as the rank problem. The problem asks, for a particular class of finitely presented groups if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank problem is one of the harder algorithmic problems studied in group theory and relatively little is known about it. Known results include:
Generalizations and related notions.
The rank of a finitely generated group "G" can be equivalently defined as the smallest cardinality of a set "X" such that there exists an onto homomorphism "F"("X") → "G", where "F"("X") is the free group with free basis "X". There is a dual notion of co-rank of a finitely generated group "G" defined as the "largest" cardinality of "X" such that there exists an onto homomorphism "G" → "F"("X"). Unlike rank, co-rank is always algorithmically computable for finitely presented groups, using the algorithm of Makanin and Razborov for solving systems of equations in free groups.
The notion of co-rank is related to the notion of a "cut number" for 3-manifolds.
If "p" is a prime number, then the "p"-rank of "G" is the largest rank of an elementary abelian "p"-subgroup. The sectional "p"-rank is the largest rank of an elementary abelian "p"-section (quotient of a subgroup).