Indecomposable Groups Research Articles

On groups with chordal power graph, including a classification in the case of finite simple groups

AbstractWe prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by (Electron J Combin 28(3):14, 2021). Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.

New examples of indecomposable torsion-free abelian groups of finite rank and rings on them

The paper deals with new specific constructions of indecomposable torsion-free abelian groups of rank two and nonzero rings on them. They illustrate purely theoretical results and complement quite rare examples obtained during the classical as well as recent research of additive groups of rings. The presented results concerning the homogeneous groups remain true for groups of any finite nonzero rank. Moreover, the paper contains a construction of a torsion-free indecomposable abelian group of an arbitrary finite rank greater than two supporting an associative, but not commutative ring, as well as a ring which is neither associative nor commutative.

A simple solution of Stratton and Webb’s problem

The paper contains a simple solution of Stratton and Webb’s problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained.

Aut-invariant quasimorphisms on free products

Let G=A *B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

Quasi-Endomorphism Rings of Some Quasi-Decomposable Torsion-Free Abelian Groups of Rank 4

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups G of rank 4, quasi-decomposable into a direct sum of groups A1 and A2 of rank 1 and a strongly indecomposable group B of rank 2 in the case where the quasi-homomorphism group ℚ ⨂ Hom(A2,B) has rank 2.

Torsion-free abelian groups revisited

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut( G ) on the set of maximal independent subsets of G determine the indecomposable decompositions. G contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.

A note on Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups in the context of additive groups of rings of finite rank

In the class of torsion-free abelian groups of finite rank, Feigelstock's conjecture on the equivalence of the notions of nil and associative nil groups is reduced to indecomposable groups. Torsion-free abelian groups $A$ of rank two such that every associative ring on $A$ is commutative but there exists a~non-commutative ring with the additive group $A$, are classified. Moreover, several valuable results concerning rings on torsion-free abelian groups of rank two achieved by Beaumont, Wisner, Jackett, Aghdam and Najafizadeh are complemented and their proof are greatly simplified.$

Indecomposable p-Local Torsion-Free Groups with Quadratic and Cubic Splitting Fields

Indecomposable torsion-free p-local Abelian groups of finite rank with quadratic and cubic splitting field K are characterized. As a consequence, for groups with quadratic splitting field K it is proved that K-decomposable p-local torsion-free Abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.

On the Quasi-Endomorphism Rings of Quasi-Decomposable Torsion-Free Abelian Groups of Rank 4 with a Strongly Indecomposable Quasi-Summand of Rank 2

We obtain a description of quasi-endomorphism rings of torsion-free Abelian groups G of rank 4 that are quasi-decomposable into a direct sum of groups A1 and A2 of rank 1 and a strongly indecomposable group B of rank 2 in the case where the quasi-homomorphism group ℚ ⨂ Hom(B,A2) has rank 2.

Finite groups with a trivial Chermak–Delgado subgroup

AbstractThe Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups ofG. The least element of the Chermak–Delgado lattice ofGis known as the Chermak–Delgado subgroup ofG. This paper concerns groups with a trivial Chermak–Delgado subgroup. We prove that if the Chermak–Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak–Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak–Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak–Delgado subgroup. We establish lattice theoretic properties of Chermak–Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichainp-group lattices play a major role in the author’s constructions.

On the Loewy length of the center of a block with elementary abelian defect groups

ABSTRACTLet G be a finite group and k an algebraically closed field of characteristic p. In this paper we investigate the Loewy structure of centers of indecomposable group algebras kG, for groups G with a normal elementary abelian Sylow p-subgroup. Furthermore, we show a reduction result for the case that a normal abelian Sylow p-subgroup is acted upon by a subgroup of its automorphism group; this is fundamental in providing generic formulae for the Loewy lengths considered.

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups $\mathcal {N}(G)$ has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.

Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A1, A2 of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ⊗ Hom(Ai, B) and Q ⊗ Hom(B, Ai) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.

Classification of a class of torsion-free abelian groups

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.

The Algebraic Structure of Finite Metabelian Group Algebras

An algorithm for the explicit computation of a complete set of primitive central idempotents, Wedderburn decomposition, and the automorphism group of the semisimple group algebra of a finite metabelian group is developed. The algorithm is illustrated with its application to the semisimple group algebra of an arbitrary metacyclic group, and certain indecomposable groups whose central quotient is the Klein four-group.

Realizations of rotations on an indecomposable compact monothetic group

By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.

A condition of indecomposability for Butler $$\mathrm B (n)$$ B ( n ) -groups

We give a manageable sufficient condition for indecomposability of Butler \(\mathrm B (n)\)-groups, allowing the easy construction of a big family of indecomposable torsionfree Abelian groups of finite rank.

Direct products and elementary equivalence of polycyclic-by-finite groups

Generalizing previous results, we give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. We use this characterization to investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular, we prove that the elementary equivalence G≡H of two such groups G, H is equivalent to each of the following properties: (1) G×⋯×G(k times G)≡H×⋯×H(k times H) for an integer k≥1; (2) A×G≡B×H for two polycyclic-by-finite groups A, B such that A≡B. It is not presently known if (1) implies G≡H for any groups G, H.

Quasi-Endomorphism Rings of Strongly Indecomposable Torsion-Free Abelian Groups of Rank 4 With Pseudosocles of Rank 1

We obtain the matrix representation of the minimal rational algebra whose subalgebras, up to isomorphism, are quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with pseudosocles of rank 1.

Representations of Posets and Indecomposable Torsion-Free Abelian Groups

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.