Finally created group

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

definition

A group is called finitely generated (or also: finitely producible ) if there is a finite subset that generates. This means that is the smallest subgroup of that contains. The subset is called the generating system of . ${\displaystyle G}$ ${\displaystyle S\subset G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle G}$

Remarks

• With you often write down the group created by. However, the generating system of a finitely generated group is not unique. ${\displaystyle \langle S\rangle }$ ${\displaystyle S}$
• In algebra, one particularly looks at finitely generated Abelian groups , since these can be classified quite easily.
• In particular, the finite groups are finitely generated, the finiteness of the group is sufficient for its finite generation, but not necessary.
• For finite producibility it is necessary that the group is a countable set . But this is not sufficient.

Examples and counterexamples

• The whole numbers are a finitely generated group with a generating system . ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle \{1,-1\}}$
• More generally, all cyclic groups are finitely generated groups.
• With the multiplication, the set of positive rational numbers forms a group that has no finite generating system, that is, it cannot be generated finitely. A minimal generating system of this group forms the countable set of prime numbers . ${\displaystyle (\mathbb {Q} ^{+},\cdot )}$
• Every free group over a finite, at least two-element set S is not commutative, finitely generated - S is a generating system - and countably infinite.

literature

• Serge Lang : Algebra (= Graduate Texts in Mathematics. Vol. 211). Revised 3rd edition. Springer, New York NY et al. 2002, ISBN 978-0-387-95385-4 .