Finite abelian groups amin witno abstract we detail the proof of the fundamental theorem of nite abelian groups, which states that every nite abelian group is isomorphic to the direct product of a unique collection of cyclic groups of prime power orders. Completely decomposable direct summands of torsionfree abelian. In a similar fashion, if r is any commutative ring, the endomorphism monoids of its modules form algebras over r by the same axioms and derivation. Classification of indecomposable abelian v, 5groups. May, 2015 though progress has been made, the question remains open. On the indecomposable torsionfree abelian groups of rank two aghdam, a. All finite abelian groups decompose into a product of not just cyclic factors, but of indecomposable cyclic factors. Because there is a unique finite field corresponding to each prime powerpr. G is viewed as an additive subgroup of a finite dimensional qvector space and its rank, rkg, is the dimension of the subspace qg that g generates. Units in abelian group algebras over direct products of. We investigate the structure of indecomposable torsionfree abelian groups all of whose pbasic subgroups are cyclic, and also the structure of the groups and rings of endomorphisms of such groups. Indecomposable representations of noncyclic groups.
Representations of posets and indecomposable torsionfree. The material on free groups, free products, and presentations of groups in terms of generators and relations see earlier handout on describing. While countable torsion abelian groups are well understood through simple presentations and ulm invariants, the case of countable mixed groups is much less mature. Strongly indecomposable finite groups sciencedirect. Finite abelian groups remain a topic of research in computational group theory. A group is called et,indecomposable if it does not have a decomposition into a direct sum of n, nonzero summands. Units in abelian group algebras over indecomposable rings peter danchev abstract. We provide an answer to this question in the case of indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic. An indecomposable abelian group g is either torsion or torsion. Let r be a commutative unitary ring of prime characteristic p which is a direct product of indecomposable subrings and let g be a multiplicative abelian group such that g0gp is nite. Every abelian groups can be isomorphically imbedded in some divisible abelian group. There exist absolutely indecomposable abelian groups up to continuous rank baer and there exist indecomposable but not absolutely indecomposable groups for all ranks r satisfying 3 r. In this section, we introduce a process to build new bigger groups from known groups. Additive notation is of course normally employed for this group.
Notes on essentially finitely indecomposable nonthick. The discussion then turns to direct sums of cyclic groups, divisible groups, and direct summands and pure subgroups, as well as kulikovs basic subgroups. In section 7 we construct new types of absolutely indecomposable groups up to at most countable rank. Motivated by examples in infinite group theory, we classify the finite.
Many mild extensions of the first order theory of abelian groups are known to be undecidable. The only indecomposable torsion groups are the cocyclic groups zpk for 1 k 1. In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation, or irrep of an algebraic structure is a nonzero representation that has no proper subrepresentation. Gr 27 feb 2018 indecomposable decompositions of torsionfree abelian groups adolf mader and phill schultz abstract. These discoveries suggested that the approach to classi cation of abelian groups up to isomorphism via direct decomposition has rather signi cant limita. These keywords were added by machine and not by the authors. Countable k,indecomposable mixed abelian groups of finite.
The class of groups under consideration is rather broad. All groups in this article are torsionfree abelian groups of finite rank. With this terminology, the cyclic groups of prime power. I want to show that as an abelian group or as an module is indecomposable. From the structure theorem for finite abelian groups, a finite cyclic group is indecomposable if and only if it is trivial or has prime power order. Home about us subjects contacts advanced search help. This seems like the perfect analogy to unique factorization in z. Since the case of cocyclic groups so of torsion groups is wellknown, in this paper we will present certain classes of torsionfree abelian groups which have indecomposable subgroups, properly constructing these subgroups. The automorphism group of a direct product of abelian groups is isomorphic to a matrix group. Classification of indecomposable abelian groups and direct. Namely, for every n 2, we construct a rank n indecomposable torsionfree abelian group which has an fa presentation. A central theme in the study of abelian groups has been the search. Fuchs problem for indecomposable abelian groups request pdf. Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.
We show that a wellknown theorem of baer and levi concerning the impossibility of simultaneous decomposition of a group into a free product and a direct sum has an analogue for abelian lattice ordered groups. An indecomposable abelian group which is not torsionfree is either a. Find all abelian groups, up to isomorphism, of order 360. We show that there exists an indecomposable module of constant jordan type 1c2d and loewy length 2 if and only if q. The proceedings of the 1987 perth conference on abelian group theory was held at the university of western australia, perth, australia on august 914, 1987. Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsionfree abelian groups. The rank of a group a is the dimension of the vector space qa that a.
Subsequent chapters focus on the structure theory of the three main classes of abelian groups. The indecomposable finite abelian groups are precisely the cyclic groups of order pa, where p is prime, a. Sep 23, 2014 if you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. Fuchs problem for indecomposable abelian groups sciencedirect. Indecomposable groups that are periodic all elements of finite orde. An abelian pgroup c is said to be essentially finitely indecomposable efi if given any decomposition of g as the direct sum of a family of subgroups, there exists a. There exists an et, indecomposable abelian group g with the torsion part t and gt% r if and only if a basic subgroup b of t can be decomposed into. Large indecomposable minimal groups large indecomposable minimal groups shelah, saharon. In the third column, we give an example of such a ring. A t where each a i is indecomposable of rank r i, so that.
This work is devoted to the study of k,indecomposable countable mixed abelian groups of finite tor sionfree rank in the most general setting. The divisible abelian groups and only they are the injective objects in the category of abelian groups. Let a, be an indecomposable polarized abelian variety. For example, zpn, where p is prime, is indecomposable but it is not simple when n 1. We recall that a group is said to be indecomposable if each two of its proper subgroups generate a proper subgroup. Mar 01, 2008 read classification of indecomposable abelian v, 5groups, discrete mathematics and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips.
Then p n is also a commutative indecomposable unitary ring. The group g may have indecomposable decompositions that result in different partitions of n. All the groups considered in this paper belong to the class g of finite rank torsionfree abelian groups. H as illustrated by considering indecomposable z4 and normal subgroup z2. This strengthens recent results due to mollovnachev commun. In the present article we describe periodic twostep solvable indecomposable groups see 1, question 7. Recall that lagranges theorem implies that the order of a subgroup must divide the order of the group. We give new examples of fa presentable torsionfree abelian groups. Home about us subject areas contacts advanced search help. On a recent test, i marked this true and found that i was incorrect.
Among finite abelian groups, only the cyclic groups are indecomposable. These discoveries suggested that the approach to classi cation of abelian groups up to isomorphism via direct decomposition has rather signi cant limitations, see the chapter on indecomposable groups in 21 for a discussion. Suppose r is a commutative indecomposable unitary ring of prime characteristic p and g is a multiplicative abelian group such that g 0g p is. A finite rank torsion free abelian group g is almost completely decom posable if there is a completely decomposable subgroup c.
Abelian groups a such that the category of asolvable groups is. Abelian group these keywords were added by machine and not by the authors. Using this as a tool we study generic abelian crossed. Indecomposable jordan types of loewy length 2 daniel bissinger abstract. It is known that there are non abelian v, 5 groups. An indecomposable decomposition of a torsionfree abelian group g of rank n is a decomposition ga1at where each ai is indecomposable of rank ri, s. Abelian group 1 abelian group in abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order the axiom of commutativity. We will study a large class of not necessarily abelian groups which can be decomposed into a direct product of indecomposable groups. As per the title of the question, i am trying to disprove the above statement.
A divisible abelian group is a direct summand of each abelian group containing it. We also cannot say that if h is a normal subgroup of g, then g. Units in abelian group algebras over indecomposable rings. It contains, for example, the groups constructed by heineken and meldrum, but, as is wellknown 36, there is an uncountable pairwise nonisomorphic family of such groups. The rst issue we shall address is the order of a product of two elements of nite order. This is to say that there are indecomposable almostfree torsionfree abelian groups. There is no other wellknown class of indecomposable torsionfree abelian groups. In the 1970s, fuchs and others discovered pathological decompositions and proved the existence of \large indecomposable groups 21. In the previous section, we took given groups and explored the existence of subgroups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class. Since the case of cocyclic groups so of torsion groups is wellknown, in this paper we will present certain classes of torsionfree abelian groups which have indecomposable subgroups, properly. We characterize the isomorphism class of the unit group urg of the group algebra rg.
A group g is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Classification of indecomposable abelian v, 5 groups classification of indecomposable abelian v, 5 groups frolov, a. Notes on essentially finitely indecomposable nonthick primary abelian groups article in communications in algebra 364. The following is a complete list of the indecomposable abelian groups which are realizable as the group of units for a ring r with c char r.
Thus our restriction of torsionfree rank to be finite is only natural. An abelian group is the same thing as a module over the ring of integers, which is the initial ring. Surface links with free abelian groups nakamura, inasa, journal of the mathematical society of japan, 2014 platonic triangles of groups alperin, roger c. Direct products and finitely generated abelian groups note. If you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. If the restriction of to an abelian subvariety a 1 of a. Notes on essentially finitely indecomposable nonthick primary. In the general case, the remakkrullshcmidt theorem theorem 3 tells us that the factorization of a group as a direct product of indecomposable groups is \unique in the sense. Units in abelian group algebras over direct products of indecomposable rings peter danchev, general kutuzov str. Torsionfree abelian groups with cyclic pbasic subgroups. The decomposability problem for torsionfree abelian. Essentially indecomposable abelian p groups having a filtration of prescribed type. Let r be a commutative unitary ring and g an abelian group.
An element of g of an abelian group g is said to be divisible in g by a positive. Now we have all the ingredients necessary to prove the following main result, which is in the focus of our investigation. This process is experimental and the keywords may be updated as the learning algorithm improves. The decomposability problem for torsionfree abelian groups. Theorem 5 from 3 showed that if g is a countable k, indecomposable abelian group, then already some finite torsionfree rank subgroup h of g with the same torsion part as g is n, indecomposable. Every finitelygenerated abelian group is a direct sum of finitely many indecomposable abelian groups. An module is said to be indecomposable if it cannot be written as a direct sum.
We describe up to isomorphism the unit group urg of the group algebra rg. On groups with finitely many indecomposable integral representations heller, a. Surface links with free abelian groups nakamura, inasa, journal of the mathematical society of japan, 2014. Indecomposable decompositions of torsionfree abelian groups. We brie y discuss some consequences of this theorem, including the classi cation of nite. Then, a decomposition of m is a finite sequence of indecomposable groups. We prove the existence of a torsionfree abelian group of countable rank with cyclic pbasic subgroups which has no indecomposable nonzero direct. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of warfield groups. Reducible divisors in indecomposable polarizations we gather in this section elementary results about divisors in polarized abelian varieties that are either nonprime or have components with high multiplicities. Constructing an indecomposable group groups of finite rank groups of in nite rank the decomposability problem for torsionfree abelian groups is analyticcomplete kyle riggs indiana university april 29, 2014 kyle riggs decomposability problem 1 22. A finitelygenerated abelian group is indecomposable if and only if it is isomorphic to z or to a factor group of the form for some prime number p and some positive integer n. Almost completely decomposable torsion free abelian groups jstor.
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