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A subgroup of a finitely generated group need not be finitely generated. The commutator subgroup of the free group on two generators is an example of a subgroup of a finitely generated group that is not finitely generated. On the other hand, all subgroups of a finitely generated abelian group are finitely generated.
However, every subgroup of a finitely generated abelian group is in itself finitely generated. In fact, more can be said: the class of all finitely generated groups is closed under extensions. To see this, take a generating set for the (finitely generated) normal subgroup and quotient. Then the generators for the normal subgroup, together with ...
Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices. [1]
In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two finitely generated subgroups of this group is again finitely generated.
For, if is finitely generated by a set , it is a quotient of the free abelian group over by a free abelian subgroup, the subgroup generated by the relators of the presentation of . But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators over B {\displaystyle B} ) forms a finite ...
lattice of subgroups The lattice of subgroups of a group is the lattice defined by its subgroups, partially ordered by set inclusion. locally cyclic group A group is locally cyclic if every finitely generated subgroup is cyclic. Every cyclic group is locally cyclic, and every finitely-generated locally cyclic group is cyclic.
Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary , there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated ...
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.