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Construction of Finite Groups

H.U. Besche, B. Eick
1999 Journal of symbolic computation  
The third method is a 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.  ... 
doi:10.1006/jsco.1998.0258 fatcat:limqcxsqanfrplrnd6rzgo3k44

Dimension of elementary amenable groups [article]

M. R. Bridson, P. H. Kropholler
2013 arXiv   pre-print
In Part I this is proved 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  ... 
arXiv:1208.1084v3 fatcat:sw3fo3ekwrgv3oa4pnrb3yguiu

On supersolubility in some groups with finitely generated Fitting radical

James C. Beidleman, Howard Smith
1992 Proceedings of the American Mathematical Society  
The main result is a supersolubility criterion 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.  ... 
doi:10.1090/s0002-9939-1992-1069288-6 fatcat:hpszei3fybhptd5mnl2e6ntqn4

On Supersolubility in Some Groups with Finitely Generated Fitting Radical

James C. Beidleman, Howard Smith
1992 Proceedings of the American Mathematical Society  
The main result is a supersolubility criterion 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.  ... 
doi:10.2307/2159649 fatcat:elzr6os3oneqfpqhsmpb4umepm

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.  ... 

On groups of finite upper rank [article]

Dan Segal
2021 arXiv   pre-print
Here we discuss the question: if the upper p-ranks of a 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.  ... 
arXiv:2104.12281v1 fatcat:2zinn63psfbnxfgp3jd6r7ctte

Dimension of elementary amenable groups

Martin R. Bridson, Peter H. Kropholler
2015 Journal für die Reine und Angewandte Mathematik  
It is conjectured that 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.  ... 
doi:10.1515/crelle-2013-0012 fatcat:5w4gfxwoarebrjg56b3ccfqf2y

POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

VITALY ROMAN'KOV
2017 Glasgow Mathematical Journal  
Let G be a 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.  ... 
doi:10.1017/s0017089516000677 fatcat:jjde5ueqinhnpmk55gxnrqx4ny

Torsion-free soluble groups, completions, and the zero divisor conjecture

W.W. Crawley-Boevey, P.H. Khopholler, P.A. Linnell
1988 Journal of Pure and Applied Algebra  
The first, proved using commutative algebra, asserts that a 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.  ... 
doi:10.1016/0022-4049(88)90029-1 fatcat:4whekt5qg5e7tijbxgcelzcg4y

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.  ... 

Groups with identities

Francçoise Point
1989 Annals of Pure and Applied Logic  
Milnor and Wolf have shown that a 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.  ... 
doi:10.1016/0168-0072(89)90060-2 fatcat:2j6yfeugffbt5k7ezha4gpas4q

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).  ... 

The soluble subgroups and the Tits alternative in linear groups over rings of fractions of polycylic group rings, I

A.I. Lichtman
1993 Journal of Pure and Applied Algebra  
Let KH be a 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  ... 
doi:10.1016/0022-4049(93)90105-3 fatcat:plbkl7sd2bcv5jw646o3medfhy

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.  ... 

Virtual rational Betti numbers of nilpotent-by-abelian groups

Behrooz Mirzaii, Fatemeh Mokari
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 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.  ... 
doi:10.2140/pjm.2016.283.381 fatcat:ty4xvko24zhsvbd6kzmj2b2tdi
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