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

${\displaystyle \operatorname {rank} (G)=\min\{|X|:X\subseteq G,\langle X\rangle =G\}.}$

If G is a finitely generated group, then the rank of G is a non-negative 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:

${\displaystyle \operatorname {sr} (G)=\max _{H\leq G}\min\{|X|:X\subseteq H,\langle X\rangle =H\}.}$

Sometimes the subgroup rank is restricted to abelian subgroups.