Homological methods and the third dimension subgroup

Int. J. Algebra Comput.

2010

Certain subquotients of group algebras are determined as a basis for subsequent computations of relative Fox and dimension subgroups. More precisely, for a group G and N-series G of G let I n R,G (G), n ≥ 0, denote the filtration of the group algebra R(G) induced by G , and I R (G) its augmentation ideal. For subgroups H of G, left ideals J of R(H) and right H -submodules M of I Z Z (G) the quotients I R (G)J/M J are studied by homological methods, notably for M = I R (G

Section: Theorem 36 Diagram (24) Is a Pushout Square Of Abelian Groumentioning

confidence: 94%

Section: The First Two Generalized Fox Quotientsmentioning

confidence: 99%

Section: Corollary 34 If the Sequencementioning

confidence: 99%

“…where a, b ∈ G, k ∈ Z Z such that a k ∈ G (2) and b k ∈ G 2 , and as usual,(aG (2) ) 2 = (aG (2) ). (aG (2) ), same for (bG (2) ) 2 ;…”

Section: The Third Fox Quotientmentioning

confidence: 99%

“…As to the second summand of ψ , let gG (2) , k, hH 2 be a typical generator of Tor Z Z 1 (G AB , H ab ) with g ∈ G, h ∈ H , k ∈ Z Z such that g k ∈ G (2) and h k ∈ H 2 .…”

Section: The Third Fox Quotientmentioning

confidence: 99%

See 3 more Smart Citations

On Fox Quotients of Arbitrary Group Algebras

Int. J. Algebra Comput.

2010

Polynomiality Properties of Group Extensions with a Torsion-Free Abelian Kernel

1996

Journal of Algebra

In studying torsion-free nilpotent groups of class 2 it is a key fact that each central group extension ‫ޚ‬ n u E¸‫ޚ‬ m is representable by a bilinear cocycle. So for investigating nilpotent groups of higher class it is natural to ask for a generalization of this fact, namely when ‫ޚ‬ m is replaced by some torsion-free nilpotent group G and ‫ޚ‬ n by some torsion-free abelian group B with nilpotent Ž . Ž G-action. In this paper we study a suitable notion of bi polynomial cocycles in the . strong sense of polynomiality introduced by Passi and prove the desired repre-Ž . sentability theorem 4.3 . This was known before only for central extensions with di¨isible kernel and with kernel ‫ޚ‬ if G is abelian, nilpotent of class 2 or if the quotients of the lower central series of G are torsion-free. Our representability 2 Ž . result implies a con¨ergence theorem for an approximation of H G, B by polyno-Ž . mial cohomology groups 4.4 . Since the latter ones are well accessible to computa-2 Ž . tion one obtains a formula for H G, B in terms of a presentation of G which can be evaluated by integral matrix calculus. More precisely, a given presentation of G amounts to a three-term cochain complex consisting of finitely generated free ‫-ޚ‬modules if the groups G and B are finitely generated; the cohomology of this 2 Ž . complex is identified with H G, B in such a way that representing 2-cocycles are Ž . explicitly given in terms of integer valued rational polynomial functions 6.4 . As an application, we establish an explicit bijection between torsion-free nilpotent groups and ‫-ޚ‬torsion-free nilpotent Lie rings both finitely generated and nilpotent of class Ž . F3 7.2 . In case a given group extension is representable by a polynomial cocycle we also determine the minimal degree of polynomiality for which this holds. Indeed, dropping the assumption that G is torsion-free nilpotent, we give an intrinsic characterization of all group extensions with torsion-free abelian kernel which are representable by a polynomial cocycle of degree F n, provided that the cokernel is EXTENSIONS WITH TORSION-FREE KERNEL 381Ž . finitely generated and acts nilpotently on the kernel 4.2 . Thus a polynomiality theory for group extensions as asked for by Passi is now achieved in case the kernel 2 Ž . is torsion-free. A motivation for thisᎏapart from calculating H G, B explicitlyᎏcomes from a question posed by J. Milnor in 1977, namely whether all finitely generated, torsion-free virtually polycyclic groups arise as fundamental groups of compact, complete affinely flat manifolds. Even the case of torsion-free nilpotent groups is interesting but far from being understood, notably since counterexamples of this type have recently been discovered. A connection of Milnor's question for nilpotent groups with polynomial constructions was first indicated in the work of P. Igodt and K. B. Lee. Recently a very close connection of this type was established, and an obstruction theory developed for the problem in terms of polynomial cohomology. On the other hand,...

The Nonabelian Tensor Square and Schur Multiplicator of Nilpotent Groups of Class 2

1996

Journal of Algebra

The nonabelian tensor square and the Schur multiplicator are determined for arbitrary groups of class 2 in a closed form. A functorial description is given in terms of a polynomial quotient of the integral group ring, as well as a more explicit formula for finite groups which can be evaluated by matrix calculus. This is carried out for 2-generator groups. Also some homotopical data of the suspended classifying space of class 2 groups are derived. The approach is based on the use of polynomial maps and constructions in the sense of Passi.