scholarly journals ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250030
Author(s):  
LUCAS SABALKA ◽  
DMYTRO SAVCHUK
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Relatively Free Group

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.

scholarly journals On torsion-free hypercentral groups with all subgroups subnormal

1989 ◽  
Vol 31 (2) ◽  
pp. 193-194
Author(s):  
Howard Smith
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Free Solvable Group ◽  
Published Result ◽  
Positive Results ◽  
Torsion Free

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

MALNORMALITY IS DECIDABLE IN FREE GROUPS

1999 ◽  
Vol 09 (06) ◽  
pp. 687-692 ◽  
Author(s):  
GILBERT BAUMSLAG ◽  
ALEXEI MYASNIKOV ◽  
VLADIMIR REMESLENNIKOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated

We prove here that there is an algorithm whereby one can decide whether or not any finitely generated subgroup of a finitely generated free group is malnormal.

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

scholarly journals AUTOMORPHIC ORBITS IN FREE GROUPS: WORDS VERSUS SUBGROUPS

2010 ◽  
Vol 20 (04) ◽  
pp. 561-590 ◽  
Author(s):  
PEDRO V. SILVA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Group Of Automorphisms ◽  
Rational Subset ◽  
Rank 2 ◽  
Higher Rank

We show that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists if we allow H to be a rational subset of F2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

scholarly journals On finitely generated submonoids of virtually free groups

2018 ◽  
Vol 10 (2) ◽  
pp. 63-82
Author(s):  
Pedro V. Silva ◽  
Alexander Zakharov
Keyword(s):  
Word Problem ◽  
Free Group ◽  
Free Groups ◽  
Isomorphism Problem ◽  
Geometric Characterization ◽  
Finitely Generated ◽  
Free Monoids ◽  
Virtually Free Group

AbstractWe prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

ON RANK, ROOT AND EQUATIONS IN FREE GROUPS

2001 ◽  
Vol 11 (03) ◽  
pp. 375-390
Author(s):  
AMNON ROSENMANN
Keyword(s):  
Directed Graph ◽  
Free Group ◽  
Free Groups ◽  
Maximal Rank ◽  
Systems Of Equations ◽  
Finitely Generated

Let h1, h2,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w1, w2, …, where each wi is a word in hi and possibly other hj, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that [Formula: see text], where [Formula: see text] is the subgroup generated by the roots of the elements in H. If H0 is finitely generated and the sequence of subgroups H0, H1, H2, … satisfies [Formula: see text] then the sequence stabilizes, i.e. for some m, Hi=Hi+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" [Formula: see text] the exponents ai satisfy gcd (a1,…,an)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

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