scholarly journals On Andrews–Curtis conjectures for soluble groups

2018 ◽  
Vol 28 (01) ◽  
pp. 97-113
Author(s):  
Luc Guyot
Keyword(s):  
Free Group ◽  
Soluble Group ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Finitely Generated Group

The Andrews–Curtis conjecture claims that every normally generating [Formula: see text]-tuple of a free group [Formula: see text] of rank [Formula: see text] can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing [Formula: see text] by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews–Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag–Solitar groups do not satisfy the generalized Andrews–Curtis conjecture in the sense of Burns and Macedońska.

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

T-systems of certain finite simple groups

1993 ◽  
Vol 113 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Martin J. Evans
Keyword(s):  
Free Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Number Of Generators

Let Fn be the free group of rank n freely generated by x1, x2,…, xn and write d(G) for the minimal number of generators of the finitely generated group G.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

On R0-Closed Classes, and Finitely Generated Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 176-184 ◽  
Author(s):  
Rex Dark ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Soluble Groups ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Central Factors ◽  
Subnormal Subgroups

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

scholarly journals Group algebras of some generalised soluble groups

1972 ◽  
Vol 18 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. P. Knott
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Soluble Group ◽  
Group Algebras ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Open Question ◽  
Torsion Free

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

scholarly journals On soluble groups which admit the dihedral group of order eight fixed-point-freely

1973 ◽  
Vol 8 (2) ◽  
pp. 305-312 ◽  
Author(s):  
Alan R. Camina ◽  
F. Peter Lockett
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Soluble Group ◽  
Dihedral Group ◽  
Finite Soluble Group ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Nilpotent Length

If the finite soluble group G admits the dihedral group of order eight as a fixed-point-free group of automorphisms then the nilpotent length of G is at most three.

scholarly journals FINITELY GENERATED SOLUBLE GROUPS WITH A CONDITION ON INFINITE SUBSETS

2012 ◽  
Vol 87 (1) ◽  
pp. 152-157
Author(s):  
ASADOLLAH FARAMARZI SALLES
Keyword(s):  
Soluble Group ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Infinite Set

AbstractLet G be a group. We say that G∈𝒯(∞) provided that every infinite set of elements of G contains three distinct elements x,y,z such that x≠y,[x,y,z]=1=[y,z,x]=[z,x,y]. We use this to show that for a finitely generated soluble group G, G/Z2(G) is finite if and only if G∈𝒯(∞).

scholarly journals Finitely approximable groups and actions Part II: Generic representations

2011 ◽  
Vol 76 (4) ◽  
pp. 1307-1321 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Free Group ◽  
Metric Space ◽  
Closed Subset ◽  
Finitely Generated ◽  
Generic Point ◽  
Finitely Generated Group ◽  
Isometric Actions

AbstractGiven a finitely generated group Γ we study the space Isom(Γ, ) of all actions of Γ by isometries of the rational Urysohn metric space , where Isom (Γ, ) is equipped with the topology it inherits seen as a closed subset of Isom . When Γ is the free group on n generators this space is just Isom , but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ, ), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on .

Relation modules of infinite groups, II

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Martin Evans
Keyword(s):  
Free Group ◽  
Vector Fields ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Continuous Vector ◽  
Finitely Generated Group ◽  
Number Of Generators ◽  
Free Modules

AbstractLet F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let $$\bar R$$ and $$\bar S$$ denote the associated relation modules of G. It is well known that $$\bar R \oplus (\mathbb{Z}G)^{d(G)} \cong \bar S \oplus (\mathbb{Z}G)^{d(G)}$$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations as above such that . Our approach depends on the existence of nonfree stably free modules over certain commutative rings and, in particular, on the existence of certain Hurwitz-Radon systems of matrices with integer entries discovered by Geramita and Pullman. This approach was motivated by results of Adams concerning the number of orthonormal (continuous) vector fields on spheres.

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