The problem is that you are confusing having a normal proper normal subgroup with being decomposable. Not every normal subgroup has a complement: this is not true even in abelian groups! For diagonalization to be impossible, one of two things must happen:.
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Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Are finite indecomposable groups necessarily simple? Ask Question. Asked 7 years, 10 months ago. Active 7 years, 10 months ago.
Viewed times. Arturo Magidin k 34 34 gold badges silver badges bronze badges. Jawad Jawad 75 4 4 bronze badges. Arturo Magidin Arturo Magidin k 34 34 gold badges silver badges bronze badges. Grigory M Grigory M Changing the generating set of the kernel of M is equivalent with multiplying M on the right by an unimodular matrix.
The Smith normal form of M is a matrix. The existence and the shape of the Smith normal proves that the finitely generated abelian group A is the direct sum. This is the fundamental theorem of finitely generated abelian groups.
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The existence of algorithms for Smith normal form shows that the fundamental theorem of finitely generated abelian groups is not only a theorem of abstract existence, but provides a way for computing expression of finitely generated abelian groups as direct sums. The simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of prime power orders.
Even though the decomposition is not unique, the number r , called the rank of A , and the prime powers giving the orders of finite cyclic summands are uniquely determined. By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups , i. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible Baer's criterion.
An abelian group without non-zero divisible subgroups is called reduced. An abelian group is called periodic or torsion , if every element has finite order. A direct sum of finite cyclic groups is periodic.
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Although the converse statement is not true in general, some special cases are known. These theorems were later subsumed in the Kulikov criterion. An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:. An abelian group that is neither periodic nor torsion-free is called mixed.
Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups.
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One of the most basic invariants of an infinite abelian group A is its rank : the cardinality of the maximal linearly independent subset of A. Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of Q and can be completely described. More generally, a torsion-free abelian group of finite rank r is a subgroup of Q r.
On the other hand, the group of p -adic integers Z p is a torsion-free abelian group of infinite Z -rank and the groups Z n p with different n are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups. The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before and form a foundation of the classification of more general infinite abelian groups.
Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings with nontrivial multiplication. Some important topics in this area of study are:. Many large abelian groups possess a natural topology , which turns them into topological groups. The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab , the prototype of an abelian category.
Nearly all well-known algebraic structures other than Boolean algebras are undecidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research:.
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Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem : are all Whitehead groups of infinite order also free abelian groups? In the s, Saharon Shelah proved that the Whitehead problem is:. Among mathematical adjectives derived from the proper name of a mathematician , the word "abelian" is rare in that it is often spelled with a lowercase a , rather than an uppercase A , indicating how ubiquitous the concept is in modern mathematics.
From Wikipedia, the free encyclopedia. For the group described by the archaic use of the related term "Abelian linear group", see Symplectic group. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations.
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