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Construction of Finite Groups

1999
*
Journal of symbolic computation
*

The third method is a

doi:10.1006/jsco.1998.0258
fatcat:limqcxsqanfrplrnd6rzgo3k44
*general*method to construct*finite**groups*which we use to compute insoluble*groups*. ... We introduce three practical algorithms to construct certain*finite**groups*up to isomorphism. The first one can be used to construct all*soluble**groups*of a given order. ... It remains to describe a procedure to compute a random*special**polycyclic**generating**sequence**for*a*finite**soluble**group*G. ...##
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Dimension of elementary amenable groups
[article]

2013
*
arXiv
*
pre-print

In Part I this is proved

arXiv:1208.1084v3
fatcat:sw3fo3ekwrgv3oa4pnrb3yguiu
*for*several classes, including the abelian-by-*polycyclic**groups*. ... It is conjectured that*for*every elementary amenable*group*G and every non-zero commutative ring k, the homological dimension of G over k is equal to the Hirsch length of G whenever G has no k-torsion. ... And in the*general*case we write G = G n with the G n*finitely**generated*, and consider the following short exact*sequence*of functors, which is in effect the*special*case of the standard spectral*sequence*...##
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On supersolubility in some groups with finitely generated Fitting radical

1992
*
Proceedings of the American Mathematical Society
*

The main result is a supersolubility criterion

doi:10.1090/s0002-9939-1992-1069288-6
fatcat:hpszei3fybhptd5mnl2e6ntqn4
*for**polycyclic**groups*. ... The*groups*G considered in this paper have the property that every normal nonsupersoluble subgroup of G has a*finite*, G-invariant, nonsupersoluble image. ... With reference to Theorem 1, a remark by a referee has prompted us to say a few words concerning*finitely**generated**soluble**groups*. ...##
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On Supersolubility in Some Groups with Finitely Generated Fitting Radical

1992
*
Proceedings of the American Mathematical Society
*

The main result is a supersolubility criterion

doi:10.2307/2159649
fatcat:elzr6os3oneqfpqhsmpb4umepm
*for**polycyclic**groups*. ... The*groups*G considered in this paper have the property that every normal nonsupersoluble subgroup of G has a*finite*, G-invariant, nonsupersoluble image. ... With reference to Theorem 1, a remark by a referee has prompted us to say a few words concerning*finitely**generated**soluble**groups*. ...##
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Page 3661 of Mathematical Reviews Vol. , Issue 91G
[page]

1991
*
Mathematical Reviews
*

In particular the conditions PH and PH” are equivalent

*for**finitely**generated**soluble**groups*and*for*torsion-free*groups*. The proof of Theorem 1 uses the deep result of J. C. Lennox and J. E. ... In*general*radicals behave badly with respect to direct products, a fact which led F. P. Lockett to introduce the *-construction*for*Fitting classes of*finite**soluble**groups*. ...##
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On groups of finite upper rank
[article]

2021
*
arXiv
*
pre-print

Here we discuss the question: if the upper p-ranks of a

arXiv:2104.12281v1
fatcat:2zinn63psfbnxfgp3jd6r7ctte
*finitely**generated**group*G are all*finite*, are they necessarily bounded? The case where G is a*soluble**group*is still an open problem. ... The 'upper rank' of a*group*is the supremum of the (Prüfer) ranks of its*finite*quotients, and*for*a prime p, the 'upper p-rank' is the supremum of the sectional p-ranks of those quotients. ... Theorem 8 establishes Conjecture B*for*the*special*case where Γ is abelianby-*polycyclic*. ...##
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Dimension of elementary amenable groups

2015
*
Journal für die Reine und Angewandte Mathematik
*

It is conjectured that

doi:10.1515/crelle-2013-0012
fatcat:5w4gfxwoarebrjg56b3ccfqf2y
*for*every elementary amenable*group*... Conjecture I.1 holds*for*abelian-by-*polycyclic*-by-*finite**groups*. Remark I.4.*For*any*group*G, hd k (G) is equal to the supremum of hd k (H) as H runs through the*finitely**generated*subgroups of G. ... More*generally*, suppose G = NS is the product of a normal nilpotent minimax*group*N and a*polycyclic**group*S*finitely**generated*. ...##
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POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

2017
*
Glasgow Mathematical Journal
*

Let G be a

doi:10.1017/s0017089516000677
fatcat:jjde5ueqinhnpmk55gxnrqx4ny
*polycyclic*, metabelian or*soluble*of type (FP)∞*group*such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. ... Every*soluble*biautomatic*group*is virtually abelian. ... At this section we will*specialize*to*soluble**groups*of type FP ∞ and establish: Theorem 5.5. If G is a*finitely**generated**soluble**group*of type (FP) ∞ , such that Rat(G) is a boolean algebra. ...##
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Torsion-free soluble groups, completions, and the zero divisor conjecture

1988
*
Journal of Pure and Applied Algebra
*

The first, proved using commutative algebra, asserts that a

doi:10.1016/0022-4049(88)90029-1
fatcat:4whekt5qg5e7tijbxgcelzcg4y
*finitely**generated*torsion-free metabelian-by-*finite**group*has many torsion-free quotients of*finite*rank. ... This paper contains two results which bear upon the zero-divisor conjecture*for**group*rings. ... We begin with a simple lemma which provides the link between*soluble**groups*of*finite*rank and*polycyclic*pro-p*groups*. Lemma 5.1. ...##
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Page 1706 of Mathematical Reviews Vol. , Issue 2003C
[page]

2003
*
Mathematical Reviews
*

Ali Reza Ashrafi (Kashan)
See also 11039.
20F

*Special*aspects of infinite or*finite**groups*2003c:20035 20F05 20D06 Erfanian, Ahmad (IR-MASHM; Mashhad) A note on growth*sequences*of alternating*groups*. ...*For*a*finite**group*G, and a positive integer k, denote by h(k, G) the largest integer n such that the power G* is*generated*by k elements. ...##
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Groups with identities

1989
*
Annals of Pure and Applied Logic
*

Milnor and Wolf have shown that a

doi:10.1016/0168-0072(89)90060-2
fatcat:2j6yfeugffbt5k7ezha4gpas4q
*finitely**generated**soluble**group*with polynomial growth has a normal nilpotent subgroup of*finite*index. ... We use the result of Rosenblatt that a*finitely**generated**soluble**group*which does not contain the free monoid on two*generators*is quasi-nilpotent. ... Let ii +A + G + G/A + 1 be an exact*sequence*and suppose that G/A has a normal*polycyclic**group*of length =+z and of index si; then A 5*finitely**generated*. ...##
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Page 3528 of Mathematical Reviews Vol. , Issue 97F
[page]

1997
*
Mathematical Reviews
*

Silvana Franciosi (Naples)
See also 51020.
20

*GROUP*THEORY AND*GENERALIZATIONS*20F*Special*aspects of infinite or*finite**groups*97f:20038 20F10 03D15 57M25 68Q25 Johnsgard, Karin (1-CRNL; Ithaca, NY) The ... A*group*G has*finite*(Priifer) rank r if every*finitely**generated*subgroup is an r-*generator**group*(and r is the least such integer). ...##
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The soluble subgroups and the Tits alternative in linear groups over rings of fractions of polycylic group rings, I

1993
*
Journal of Pure and Applied Algebra
*

Let KH be a

doi:10.1016/0022-4049(93)90105-3
fatcat:plbkl7sd2bcv5jw646o3medfhy
*group*ring of a*polycyclic*-by-*finite**group*and let R be its Goldie ring of fractions. ... In this first paper in a series of two we study the*soluble*subgroups of the linear*group*GL,(R) and show in particular that there exists a bound*for*their*solubility*class; we will show in the second ... Wehrfritz*for*valuable remarks which helped to avoid errors and to improve the exposition. I would like to thank Ram Murty*for*his interest in my work: his solution of a few number-theoretical ...##
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Page 336 of Mathematical Reviews Vol. 42, Issue 2
[page]

1971
*
Mathematical Reviews
*

This theorem is a consequence of Theorem A2: If a

*group*G possesses an ascendant*polycyclic*sub-*group*H with trivial centralizer, then H® is a*soluble*z- minimax*group**for*some*finite*set of primes 7 ...*For*example, Theorem D: If H is a subnormal*soluble*7-minimax subgroup of a local!y*soluble**group*G@ and if H has*finite*centralizer in G, then G@ is a*soluble*7-minimax*group*. ...##
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Virtual rational Betti numbers of nilpotent-by-abelian groups

2016
*
Pacific Journal of Mathematics
*

More precisely, we prove that if N/N' is 2(c(n-1)-1)-tame as a G/N-module, c the nilpotency class of N, then vb_j(G):=sup_M∈A_G_Q H_j(M,Q) is

doi:10.2140/pjm.2016.283.381
fatcat:ty4xvko24zhsvbd6kzmj2b2tdi
*finite**for*all 0≤ j≤ n, where A_G is the set of all*finite*... In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian*group*G, where the abelianization N/N' of its nilpotent part N satisfies certain tameness property. ... Moreover, the*special*case of Theorem 2.1*for*c = 2 also was proved by J. R. Groves which was made available to us by D. H. Kochloukova. His proves is different than ours. ...
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