We show that the class of groups where EDT0L languages can be used to describe solution sets to systems of equations is closed under direct products, wreath products with finite groups, and passing ...to finite index subgroups. We also add the class of groups that contain a direct product of hyperbolic groups as a finite index subgroup to the list of groups where solutions to systems of equations can be expressed as an EDT0L language. This includes dihedral Artin groups. We also show that the systems of equations with rational constraints in virtually abelian groups have EDT0L solutions, and the addition of recognisable contraints to any system preserves the property of having EDT0L solutions. These EDT0L solutions are expressed with respect to quasigeodesic normal forms. We discuss the space complexity in which EDT0L systems for these languages can be constructed.
The Gorenstein cohomological dimension is defined for any group G and coincides with the virtual cohomological dimension vcdG, whenever the latter is defined and finite. Unlike the virtual ...cohomological dimension, the Gorenstein cohomological dimension behaves well with respect to extensions, finite graphs of groups and ascending unions. In this paper, we study the Gorenstein cohomological dimension GcdkG of groups G, which are of type FP∞ over a commutative ring k. We show that if 1⟶N⟶G⟶Q⟶1 is an extension of groups with N of type FP∞ over a field F and GcdFQ<∞, then GcdFG=GcdFN+GcdFQ. We also show that for any group G of type FP∞ over Z with GcdZG<∞, there exists a field F such that GcdFG=GcdZG. This implies, in particular, that if G is a group of type FP∞ over Z and G is an extension of N by itself, then GcdZG=2GcdZN.
The Janko sporadic simple group J2 has an automorphism group 2. Using the electronic Atlas of Wilson 22, the group J2:2 has an absolutely irreducible module of dimension 12 over F2. It follows that a ...split extension group of the form 2^12:(J2:2) := G exists. In this article we study this group, where we compute its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. The inertia factor groups of G will be determined by analysing the maximal subgroups of J2:2 and maximal of the maximal subgroups of J2:2 together with various other information. It turns out that the character table of G is a 64×64 real valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 6.
Equations over solvable groups Klyachko, Anton A.; Mikheenko, Mikhail A.; Roman'kov, Vitaly A.
Journal of algebra,
01/2024, Letnik:
638
Journal Article
Recenzirano
Odprti dostop
Not any nonsingular equation over a metabelian group has solution in a larger metabelian group. However, any nonsingular equation over a solvable group with a subnormal series with abelian ...torsion-free quotients has a solution in a larger group with a similar subnormal series of the same length (and an analogous fact is valid for systems of equations).
On invariant (co)homology of a group Aquino, Carlos; Jimenez, Rolando; Mijangos, Martin ...
Topology and its applications,
04/2021, Letnik:
293
Journal Article
Recenzirano
Odprti dostop
There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that ...arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification of (invariant) group extensions.
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