Minimum size of generating set
This article defines an arithmetic function on groups
View other such arithmetic functions
This number is finite if and only if the group is a finitely generated group.
|Upper bound on minimum size of generating set||Name of groups satisfying this upper bound|
|2||2-generated group (examples include symmetric group on a finite set, see symmetric group on a finite set is 2-generated)|
- 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
Relation with other arithmetic functions
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.
Arithmetic functions taking values greater than or equal to minimum size of generating set
|Arithmetic function||Meaning||Proof of comparison|
|maximum size of minimal generating set||The maximum, over all minimal generating sets of the group, of their sizes|
|max-length of a group||maximum possible length of a subgroup series for the group||(via maximum size of minimal generating set)|
|sum of exponents of prime divisors in prime factorization of order||(via max-length). See minimum size of generating set is bounded by sum of exponents of prime divisors of order|