scholarly journals Torsion Z-module and Torsion-free Z-module

2014 ◽  
Vol 22 (4) ◽  
pp. 277-289
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Finitely Generated ◽  
Torsion Free

Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

scholarly journals Matrix of ℤ-module

2015 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Linear Transformation ◽  
Bilinear Form ◽  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Gramian Matrix ◽  
Selection Of

Summary In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals Divisible ℤ-modules

2016 ◽  
Vol 24 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Coefficient Ring ◽  
Torsion Free

Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

A Finite Index Property of Certain Solvable Groups

1984 ◽  
Vol 27 (4) ◽  
pp. 485-489
Author(s):  
A. H. Rhemtulla ◽  
H. Smith
Keyword(s):  
Solvable Group ◽  
Finite Index ◽  
Finite Rank ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Torsion Free

AbstractA group G is said to have the FINITE INDEX property (G is an FI-group) if, whenever H≤G, xp ∈ H for some x in G and p > 0, then |〈H, x〉: H| is finite. Following a brief discussion of some locally nilpotent groups with this property, it is shown that torsion-free solvable groups of finite rank which have the isolator property are FI-groups. It is deduced from this that a finitely generated torsion-free solvable group has an FI-subgroup of finite index if and only if it has finite rank.

scholarly journals Torsion Part of ℤ-module

2015 ◽  
Vol 23 (4) ◽  
pp. 297-307 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Rational Numbers ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Torsion Part ◽  
Torsion Free

Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

scholarly journals Soluble groups isomorphic to their non-nilpotent subgroups

1999 ◽  
Vol 67 (3) ◽  
pp. 399-411 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Torsion Free

AbstractA group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

Sign in / Sign up

Export Citation Format

Share Document