For example, the additive group z is a free abelian group. Every finitelygenerated abelian group is a direct sum of finitely many indecomposable abelian groups. The material on free groups, free products, and presentations of groups in terms of generators and relations see earlier handout on describing. 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. On groups with finitely many indecomposable integral representations heller, a. 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.
If you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. Then, a decomposition of m is a finite sequence of indecomposable groups. An indecomposable abelian group which is not torsionfree is either a. Every abelian groups can be isomorphically imbedded in some divisible abelian group. We prove the existence of a torsionfree abelian group of countable rank with cyclic pbasic subgroups which has no indecomposable nonzero direct. Units in abelian group algebras over direct products of indecomposable rings peter danchev, general kutuzov str. In this section, we introduce a process to build new bigger groups from known groups. 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. Indecomposable jordan types of loewy length 2 daniel bissinger abstract. 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. Subsequent chapters focus on the structure theory of the three main classes of 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. This is to say that there are indecomposable almostfree torsionfree abelian groups. The divisible abelian groups and only they are the injective objects in the category of abelian groups. The decomposability problem for torsionfree abelian. An module is said to be indecomposable if it cannot be written as a direct sum. Let r be a commutative unitary ring and g an abelian group. Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products. Find all abelian groups, up to isomorphism, of order 360. H as illustrated by considering indecomposable z4 and normal subgroup z2. A finite rank torsion free abelian group g is almost completely decom posable if there is a completely decomposable subgroup c. This work is devoted to the study of k,indecomposable countable mixed abelian groups of finite tor sionfree rank in the most general setting.
Surface links with free abelian groups nakamura, inasa, journal of the mathematical society of japan, 2014. 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. The group g may have indecomposable decompositions that result in different partitions of n. 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. Classification of indecomposable abelian groups and direct. We give new examples of fa presentable torsionfree abelian groups. With this terminology, the cyclic groups of prime power. Fuchs problem for indecomposable abelian groups sciencedirect.
Because there is a unique finite field corresponding to each prime powerpr. In the present article we describe periodic twostep solvable indecomposable groups see 1, question 7. An abelian group is the same thing as a module over the ring of integers, which is the initial ring. Wilkins academic year 19967 6 groups a binary operation. 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. An indecomposable decomposition of a torsionfree abelian group g of rank n is a decomposition g a 1. In the third column, we give an example of such a ring.
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. Namely, for every n 2, we construct a rank n indecomposable torsionfree abelian group which has an fa presentation. A note on the square subgroups of decomposable torsion. Now we have all the ingredients necessary to prove the following main result, which is in the focus of our investigation. 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. A t where each a i is indecomposable of rank r i, so that. Fuchs problem for indecomposable abelian groups request pdf. All the groups considered in this paper belong to the class g of finite rank torsionfree abelian groups. Motivated by examples in infinite group theory, we classify the finite.
We brie y discuss some consequences of this theorem, including the classi cation of nite. 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. Notes on essentially finitely indecomposable nonthick primary. We also cannot say that if h is a normal subgroup of g, then g. A central theme in the study of abelian groups has been the search. We describe up to isomorphism the unit group urg of the group algebra rg. An element of g of an abelian group g is said to be divisible in g by a positive. May, 2015 though progress has been made, the question remains open. Classification of indecomposable abelian v, 5groups. Strongly indecomposable finite groups sciencedirect. Home about us subject areas contacts advanced search help. 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. On a recent test, i marked this true and found that i was incorrect.
The rst issue we shall address is the order of a product of two elements of nite order. Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsionfree abelian groups. Torsionfree groups of rank 1 subgroups of q are indecomposable. Abelian group these keywords were added by machine and not by the authors. Countable k,indecomposable mixed abelian groups of finite. Units in abelian group algebras over direct products of.
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. 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. Essentially indecomposable abelian p groups having a filtration of prescribed type. The indecomposable finite abelian groups are precisely the cyclic groups of order pa, where p is prime, a. Units in abelian group algebras over indecomposable rings peter danchev abstract.
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. Home about us subjects contacts advanced search help. The rank of a group a is the dimension of the vector space qa that a. All groups in this article are torsionfree abelian groups of finite rank. 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. 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. Then p n is also a commutative indecomposable unitary ring.
Large indecomposable minimal groups large indecomposable minimal groups shelah, saharon. 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. All finite abelian groups decompose into a product of not just cyclic factors, but of indecomposable cyclic factors. The discussion then turns to direct sums of cyclic groups, divisible groups, and direct summands and pure subgroups, as well as kulikovs basic subgroups. Direct products and finitely generated abelian groups note. 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. Among finite abelian groups, only the cyclic groups are indecomposable. Units in abelian group algebras over indecomposable rings. Let a, be an indecomposable polarized abelian variety. In section 7 we construct new types of absolutely indecomposable groups up to at most countable rank. In the 1970s, fuchs and others discovered pathological decompositions and proved the existence of \large indecomposable groups 21. The class of groups under consideration is rather broad. 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. Almost completely decomposable torsion free abelian groups jstor.
It is known that there are non abelian v, 5 groups. A group is called et,indecomposable if it does not have a decomposition into a direct sum of n, nonzero summands. This seems like the perfect analogy to unique factorization in z. Completely decomposable direct summands of torsionfree abelian. We show that there exists an indecomposable module of constant jordan type 1c2d and loewy length 2 if and only if q. The set of all 2dimensional vectors is an abelian group under the operation of addi tion. The automorphism group of a direct product of abelian groups is isomorphic to a matrix group. 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. 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. 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. As per the title of the question, i am trying to disprove the above statement. Notes on essentially finitely indecomposable nonthick.
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. On the indecomposable torsionfree abelian groups of rank two aghdam, a. Indecomposable representations of noncyclic groups. The only indecomposable torsion groups are the cocyclic groups zpk for 1 k 1. Gr 27 feb 2018 indecomposable decompositions of torsionfree abelian groups adolf mader and phill schultz abstract. Torsionfree abelian groups with cyclic pbasic subgroups. Abelian groups a such that the category of asolvable groups is. 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. While countable torsion abelian groups are well understood through simple presentations and ulm invariants, the case of countable mixed groups is much less mature. Sep 23, 2014 if you restrict to finite abelian groups, then all indecomposable such groups are cyclic, but not all cyclic groups are indecomposable. A divisible abelian group is a direct summand of each abelian group containing it. Additive notation is of course normally employed for this group. 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. The decomposability problem for torsionfree abelian groups.
Surface links with free abelian groups nakamura, inasa, journal of the mathematical society of japan, 2014 platonic triangles of groups alperin, roger c. These keywords were added by machine and not by the authors. Recall that lagranges theorem implies that the order of a subgroup must divide the order of the group. Classification of indecomposable abelian v, 5 groups classification of indecomposable abelian v, 5 groups frolov, a. 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.
Representations of posets and indecomposable torsionfree. 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. Indecomposable decompositions of torsionfree abelian groups. There is no other wellknown class of indecomposable torsionfree abelian groups. Using this as a tool we study generic abelian crossed. 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 group g is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. We recall that a group is said to be indecomposable if each two of its proper subgroups generate a proper subgroup. These discoveries suggested that the approach to classi cation of abelian groups up to isomorphism via direct decomposition has rather signi cant limita. Notes on essentially finitely indecomposable nonthick primary abelian groups article in communications in algebra 364. In the previous section, we took given groups and explored the existence of subgroups. Many mild extensions of the first order theory of abelian groups are known to be undecidable.
This process is experimental and the keywords may be updated as the learning algorithm improves. Indecomposable groups that are periodic all elements of finite orde. If the restriction of to an abelian subvariety a 1 of a. 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. Still, i am confused and cannot think of a counterexample. Thus our restriction of torsionfree rank to be finite is only natural. This strengthens recent results due to mollovnachev commun. Finite abelian groups remain a topic of research in computational group theory. I want to show that as an abelian group or as an module is indecomposable.
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