Finally generated Abelian group

A finitely generated Abelian group is an Abelian group that is finitely generated . The main theorem on finitely generated Abelian groups provides a complete classification of these groups. ${\displaystyle (G,+)}$

Examples and counterexamples

• All finite groups are finitely generated. Hence finite Abelian groups are also finitely generated.
• The integers are an infinite Abelian group that is finitely generated with 1 as the generator. ${\displaystyle (\mathbb {Z} ,+)}$
• Every direct sum of finitely many finitely generated Abelian groups is again a finitely generated Abelian group.
• The additive group of the rational numbers is not finitely generated: one has to choose a natural number that is relatively prime to the denominators of all ; then can not be represented as an integer linear combination of . ${\displaystyle (\mathbb {Q} ,+)}$ ${\displaystyle x_{1},\ldots ,x_{s}}$ ${\displaystyle w}$ ${\displaystyle x_{i}}$ ${\displaystyle {\tfrac {1}{w}}}$ ${\displaystyle x_{1},\ldots ,x_{s}}$

classification

Every subgroup and factor group of a finitely generated Abelian group is again finitely generated Abelian. The finitely generated Abelian groups together with the group morphisms form an Abelian category .

Note that not every Abelian group of finite rank is finitely generated. for example is of rank 1 but not finitely generated. Another example is the direct sum of infinitely many copies of , this is of rank 0, but also not finitely generated. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

The main theorem about finitely generated Abelian groups says that every finitely generated Abelian group is isomorphic to a finite direct sum of cyclic groups, whose order is the power of a prime number , and infinite cyclic groups . ${\displaystyle G}$

Finite Abelian groups

• For every natural number with the prime factorization there are exactly isomorphy types of Abelian groups with elements. The function is the partition function , the result is sequence A000688 in OEIS . ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle N={p_{1}}^{r_{1}}\cdot {p_{2}}^{r_{2}}\cdots {p_{k}}^{r_{k}}}$ ${\displaystyle a(N)=P(r_{1})\cdot P(r_{2})\cdots P(r_{k})}$ ${\displaystyle N}$ ${\displaystyle P}$ ${\displaystyle a(N)}$

• Every such Abelian group with elements has a generating system of at most elements. ${\displaystyle N}$ ${\displaystyle \max(r_{1},r_{2},\ldots r_{k})}$

• The following applies in particular: If a natural number is a square-free natural number, then every Abelian group with elements is cyclic. ${\displaystyle N}$ ${\displaystyle N}$

literature

• Thomas W. Hungerford: Algebra . In: Graduate texts in mathematics . 8th corrected edition. No. 73. Springer, New York / Berlin / Singapore / Tokyo / Heidelberg / Barcelona / Budapest / Hong Kong / London / Milan / Paris / Santa Clara 1996, ISBN 3-540-90518-9 , II. The Structure of Groups, 2. Finitely Generated Abelian Groups, p. 76–82 ( filestube.com [PDF; 8.0 MB ; accessed on February 15, 2012] DNB 949253235/04 table of contents).