Generating set of a group

A finitely generated group is an object from the mathematical branch of abstract algebra. It is a special case of a group.

Definition

A group is called finitely generated (or: finitely generated ) if there is a finite subset which generates. This means that the smallest subgroup is containing. The subset is called a system of generators.

Comments

• With one notes often the set of generated. However, the system of generators of a finitely generated group is not unique.
• In algebra, we consider in particular finitely generated abelian groups, since one can classify these fairly simple.
• The finite groups are finitely generated particular, the finiteness of the group is sufficient for their finite producibility, but not necessary.
• It is necessary for the finite producibility that the group is a countable set. But this is not sufficient!

Examples and counter-examples

• The integers are a finitely generated group with generating set.
• More generally, all cyclic groups of finitely generated groups.
• The set of positive rational numbers together with the multiplication a group that does not have a finite generating system, so do not finally be generated. A minimal system of generators of this group is the countable set of primes.
• Every free group on a finite, at least two-element set S is not commutative, finitely generated - S is a generating system - and countably infinite.