Decomposition of a specific class of (1,3) groups

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2022-12-21

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Çölaşan, Utku Şükrü

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The classification (up to near isomorphism) of some class of almost completely decomposable groups with a regulating regulator is possible. Since almost completely decomposable groups can be written as a direct sum of indecomposable groups, for classification of almost completely decomposable groups it would be enough to find isomorphism classes of all indecomposable groups. The class of almost completely decomposable groups with a critical typeset in (1,3) configuration and a regulator quotient of exponent p3 have 6 near isomorphism classes of indecomposable groups. We describe almost completely decomposable groups by coordinate matrices. If the coordinate matrix of an almost completely decomposable group is decomposable, then the group is decomposable. Hence the method used in this thesis to determine the decomposition of almost completely decomposable groups is turned to an equivalance problem of determining the decomposition of the corresponding coordinate matrices.

Subject Keywords

Almost completely decomposable groups, Torsion free groups, Decomposability of almost completely decomposable groups

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https://hdl.handle.net/11511/101279

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Graduate School of Natural and Applied Sciences, Thesis

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Almost completely decomposable groups and unbounded representation type Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2012-01-01) Almost completely decomposable groups with a regulating regulator and a p-primary regulator quotient are studied. It is shown that there are indecomposable such groups of arbitrarily large rank provided that the critical typeset contains some basic configuration and the exponent of the regulator quotient is sufficiently large.
Classification of a class of torsion-free abelian groups Solak, Ebru (2016-01-01) The class of almost completely decomposable groups with a critical typeset of type (2, 2) and a regulator quotient of exponent <= p(2) is shown to have exactly 4 near-isomorphism classes of indecomposable groups. Every group of the class is up to near-isomorphism uniquely a direct sum of these four indecomposable groups.
Indecomposable (1,3)-Groups and a matrix problem Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2013-06-01) Almost completely decomposable groups with a critical typeset of type (1, 3) and a p-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient p (k) , either no indecomposables if k a (c) 1/2 2; only six near isomorphism types of indecomposables if k = 3; and indecomposables of arbitrary large rank if k a (c) 3/4 4.
(1,4)-GROUPS WITH HOMOCYCLIC REGULATOR QUOTIENT OF EXPONENT p(3) Arnold, David M.; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2015-01-01) The class of almost completely decomposable groups with a critical typeset of type (1,4) and a homocyclic regulator quotient of exponent p(3) is shown to be of bounded representation type. There are precisely four near-isomorphism classes of indecomposables, all of rank 6.
REPRESENTATIONS OF POSETS AND INDECOMPOSABLE TORSION-FREE ABELIAN GROUPS Arnold, David; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru (2014-03-04) Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. 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.

Citation Formats

U. Ş. Çölaşan, “Decomposition of a specific class of (1,3) groups,” M.S. - Master of Science, Middle East Technical University, 2022.