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.
According to the classic Grushko theorem, rank behaves additively with respect to taking free products, that is, for any groups A and B we have
rank(AB) = rank(A) + rank(B).
If is a one-relator group such that r is not a primitive element in the free group F(x1,..., xn), that is, r does not belong to a free basis of F(x1,..., xn), then rank(G) = n.
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:
The rank problem is algorithmically undecidable for the class of all finitely presented groups. Indeed, by a classical result of Adian-Rabin, there is no algorithm to decide if a finitely presented group is trivial, so even the question of whether rank(G)=0 is undecidable for finitely presented groups.
The rank problem is decidable for finite groups and for finitely generated abelian groups.
If p is a prime number, then the p-rank of G is the largest rank of an elementary abelianp-subgroup. The sectionalp-rank is the largest rank of an elementary abelian p-section (quotient of a subgroup).
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