Minimum size of generating set

Definition
Let $$G$$ be a group. The minimum size of generating set for $$G$$, often called the rank or generating set-rank of $$G$$, and sometimes denoted $$d(G)$$ or $$r(G)$$, is defined as the minimum possible size of a generating set for $$G$$.

This number is finite if and only if the group is a finitely generated group.

Facts

 * Minimum size of generating set of subgroup may be more than of whole group
 * Minimum size of generating set of quotient is less than or equal to that of whole group
 * Cyclicity is subgroup-closed, i.e., the property of minimum size of generating set being at most 1 is closed under taking subgroups.
 * Minimum size of generating set of direct product of two groups is bounded by sum of minimum size of generating set for each

Arithmetic functions defined using it

 * Subgroup rank of a group: This is the maximum of the generating set-ranks over all subgroups of the group.
 * Rank of a p-group: For a group of prime power order, this is the maximum of the ranks of all the abelian subgroups of the group.