finitely generated group
A finitely generated group is a group that has a finite generating set.
Every finite group
is obviously finitely generated
.
Every finitely generated group is countable.
Any quotient (http://planetmath.org/QuotientGroup)
of a finitely generated group is finitely generated.
However, a finitely generated group may have subgroups
that are not finitely generated.
(For example, the free group
of rank is generated by just two elements,
but its commutator subgroup
is not finitely generated.)
Nonetheless, a subgroup of finite index in a finitely generated group
is necessarily finitely generated;
a bound on the number of generators
required for the subgroup is given by
the Schreier index formula (http://planetmath.org/ScheierIndexFormula).
The finitely generated groups
all of whose subgroups are also finitely generated
are precisely the groups satisfying the maximal condition.
This includes all finitely generated nilpotent groups and,
more generally, all polycyclic groups
.
A group that is not finitely generated is sometimes said to be infinitely generated.
| Title | finitely generated group |
| Canonical name | FinitelyGeneratedGroup |
| Date of creation | 2013-03-22 12:16:38 |
| Last modified on | 2013-03-22 12:16:38 |
| Owner | yark (2760) |
| Last modified by | yark (2760) |
| Numerical id | 24 |
| Author | yark (2760) |
| Entry type | Definition |
| Classification | msc 20A05 |
| Related topic | FundamentalTheoremOfFinitelyGeneratedAbelianGroups |
| Related topic | AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex |
| Defines | finitely generated |
| Defines | finitely generated subgroup |
| Defines | infinitely generated |
| Defines | infinitely generated group |
| Defines | infinitely generated subgroup |