Law on finitely generated Abelian groups

The main theorem about finitely generated Abelian groups is a result of group theory , especially the theory about finitely generated Abelian groups . These are groups that commute under their binary connection and in which each element can be represented as a product of elements of a finite set of producers. ${\displaystyle g\in G}$

The statement of the theorem is that for all these groups there is a breakdown or decomposition into a finite number of cyclic subgroups , that is, groups that are generated by exactly one element. The group is the direct product of these subgroups. Because every cyclic group of finite order is isomorphic to a residue class group and every cyclic group of infinite order is isomorphic to the group of integers , each of these groups is isomorphic to a product of an infinite or trivial group of the type with a finite group that is a product of Is residual class groups. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle (\mathbb {Z} /d\mathbb {Z} ,+)}$ ${\displaystyle (\mathbb {Z} ,+)}$ ${\displaystyle \mathbb {Z} ^{n}\,(n\in \mathbb {N} )}$

In other words, the main theorem says that a finitely generated Abelian group is the direct product of a free Abelian group of finite rank and a finite Abelian group. The finite Abelian group is the torsional subgroup of . The free Abelian group is generally not uniquely determined, only its rank. ${\displaystyle G}$

The theorem follows directly from the theorem about the classification of finitely generated modules over main ideal rings , since every Abelian group can be understood as a module over the main ideal ring of integers.

statement

Is an Abelian group finitely generated, so there are certain non-negative integers clearly and unambiguously certain prime powers with ${\displaystyle G}$ ${\displaystyle r,t}$ ${\displaystyle 1<d_{1}\leq \ldots \leq d_{t}}$

${\displaystyle G\cong \mathbb {Z} ^{r}\oplus \bigoplus _{i=1}^{t}\mathbb {Z} /d_{i}\mathbb {Z} .}$

Proof idea

The existence of the decomposition is shown by starting with any generating system by means of elementary transformations, constructing a suitable possibly other generating system that allows a summand to be split off. In this way a proof by full induction on the number of producers is possible.

Conclusions and Examples

• The properties free , projective , torsion-free , flat are equivalent in finitely generated Abelian groups.

For finite Abelian groups

For the isomorphy type of the cyclic group with elements, the following is abbreviated , groups are written “multiplicative”, as is customary in the theory of finite groups, and the direct sums from the main theorem are accordingly as direct products . ${\displaystyle m}$ ${\displaystyle (\mathbb {Z} /m\mathbb {Z} ,+)}$ ${\displaystyle C_{m}}$

Every finite, non-trivial, Abelian p-group ( positive prime number) has a power as a group order. For every number partition of there is exactly one Abelian group with elements except for isomorphism . The number of payment partitions can be specified with the partition function . ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle p^{n},n\in \mathbb {N} .}$ ${\displaystyle n}$ ${\displaystyle p^{n}}$ ${\displaystyle P(n)}$

example

• Exactly different types of isomorphism of Abelian groups exist for the group order ${\displaystyle 16=2^{4}}$ ${\displaystyle P(4)=5}$

${\displaystyle {C_{2}}^{4}}$to partition 4 = 1 + 1 + 1 + 1, to partition 4 = 2 + 1 + 1, to partition 4 = 2 + 2, to partition 4 = 3 + 1 and to partition 4 = 4. ${\displaystyle C_{4}\times {C_{2}}^{2}}$ ${\displaystyle {C_{4}}^{2}}$ ${\displaystyle C_{8}\times C_{2}}$ ${\displaystyle C_{16}}$

Together with the statement from elementary number theory if and only if are coprime , we get: ${\displaystyle C_{r}\times C_{s}\cong C_{r\cdot s}}$ ${\displaystyle r,s}$

• Exactly when is a square-free natural number, that is, when the square is not a factor of any prime number , there is only one Abelian group with elements apart from isomorphism . The group is then cyclical and it applies ${\displaystyle N}$ ${\displaystyle p}$ ${\displaystyle p^{2}}$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle G\cong C_{N}}$
• If the prime factorization is , then there exist exactly Abelian groups with elements except for isomorphism . ( etc. is the partition function.) ${\displaystyle N={p_{1}}^{r_{1}}\cdot {p_{2}}^{r_{2}}\cdot \cdots \cdot {p_{k}}^{r_{k}}}$ ${\displaystyle N}$ ${\displaystyle P(r_{1})\cdot P(r_{2})\cdot \cdots \cdot P(r_{k})}$ ${\displaystyle N}$ ${\displaystyle P(r_{1})}$

Each such group has a generating system of at most group elements. ${\displaystyle R:=\max(r_{1},r_{2},\ldots r_{k})}$

• Every finite Abelian group with the group order is isomorphic to a direct product ${\displaystyle G}$ ${\displaystyle N}$

${\displaystyle C_{m_{1}}\times C_{m_{2}}\times \cdots \times C_{m_{s}},}$applies , always divides for and applies to the product of all these numbers . ${\displaystyle m_{1}>1}$ ${\displaystyle m_{k}}$ ${\displaystyle m_{k+1}}$ ${\displaystyle 1\leq k<s}$ ${\displaystyle m_{1}\cdot m_{2}\cdot \cdots m_{s}=N}$

• The specified product presentation is clearly determined by the group and the divisibility requirement. ${\displaystyle G}$
• The maximum order of a group element , for all group elements applies , and any other natural number , for all group elements is considered, is a multiple of . ${\displaystyle M=m_{s}}$ ${\displaystyle g\in G}$ ${\displaystyle g^{M}=e}$ ${\displaystyle K}$ ${\displaystyle g^{K}=e}$ ${\displaystyle g\in G}$ ${\displaystyle M}$
• The group has a generating system of group elements, and each generating system contains at least elements. In this respect, the representation given is a “minimal product representation” of the group. ${\displaystyle s}$ ${\displaystyle s}$

Examples

• The Abelian group has the isomorphism type as shown in the main clause. With the help of a sorted table of the orders of prime numbers that occur, one obtains the mentioned "minimal product representation": ${\displaystyle G}$ ${\displaystyle G\cong {C_{2}}^{2}\times C_{3}\times C_{4}\times {C_{9}}^{2}\times {C_{27}}^{2}}$

Powers of 3:	3	9	9	27	27
Powers of 2:	1	1	2	2	4th
Products:	3	9	18th	54	108

You sort according to the ascending exponents of the prime power and fill in lines that contain less than 5 powers with 1 from the beginning. The last line, which contains the products of the columns, then contains the ascending chain of factors. The results , as shown mentioned with increasing dividers, so to get this group from a generating set of five group members - 5 is the maximum number of p-groups to a prime number that occur in the product presentation in accordance with the law! ${\displaystyle G\cong C_{3}\times C_{9}\times C_{18}\times C_{54}\times C_{108}}$

• For the Abelian group ${\displaystyle G=C_{3}\times C_{6}\times C_{6}\times C_{60}}$

Products:	3	6th	6th	60
Powers of 2:	1	2	2	4th
Powers of 3:	3	3	3	3
Powers of 5:	1	1	1	5

one first tabulates the ascending divisors, factoring them according to the occurring prime powers and thus obtaining the representation according to the main theorem . A minimal generating system of this group contains four elements. ${\displaystyle G\cong {C_{2}}^{2}\times {C_{3}}^{4}\times C_{4}\times C_{5}}$

literature

• Müller-Stach, Piontkowski: Elementary and algebraic number theory. Vieweg, ISBN 978-3-8348-0211-8 .
• Leutbecher: Number Theory - An Introduction to Algebra. Springer, ISBN 978-3-540-58791-0 .
• Siegfried Bosch : Algebra. 7th edition. Springer-Verlag, 2009, ISBN 3-540-40388-4 , doi: 10.1007 / 978-3-540-92812-6 .

Remarks

1. ↑ In this article, the operation is seen as multiplicative. It is only a notation and one could easily speak of multiples . This is no longer indicated below.

Web links

Wikibooks: Proof of the Theorem  - Learning and Teaching Materials