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.
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.
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.
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
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 .
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 .
- example
- Exactly different types of isomorphism of Abelian groups exist for the group order
- 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.
Together with the statement from elementary number theory if and only if are coprime , we get:
- 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
- If the prime factorization is , then there exist exactly Abelian groups with elements except for isomorphism . ( etc. is the partition function.)
- Each such group has a generating system of at most group elements.
- Every finite Abelian group with the group order is isomorphic to a direct product
-
applies , always divides for and applies to the product of all these numbers .
- The specified product presentation is clearly determined by the group and the divisibility requirement.
- 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 .
- 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.
- 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":
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!
- For the Abelian group
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.
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
- ↑ 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.