Inverse zero-sum problems and algebraic invariants
Inverse zero-sum problems in finite Abelian p-groups
We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...
Isolated subgroups of finite abelian groups
We say that a subgroup is isolated in a group if for every we have either or . We describe the set of isolated subgroups of a finite abelian group. The technique used is based on an interesting connection between isolated subgroups and the function sum of element orders of a finite group.
Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order
In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].
Isomorphisms of Direct Products of Finite Commutative Groups
We have been working on the formalization of groups. In [1], we encoded some theorems concerning the product of cyclic groups. In this article, we present the generalized formalization of [1]. First, we show that every finite commutative group which order is composite number is isomorphic to a direct product of finite commutative groups which orders are relatively prime. Next, we describe finite direct products of finite commutative groups
Isomorphisms of Direct Products of Finite Cyclic Groups
In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.
Konstruktion aller Automorphismen der Ordnung 2 einer endlichen elementaren abelschen Gruppe
Maximal sum-free sets and block designs
Minimum sum and difference covers of abelian groups.
Multiple Tilings of n-Dimensional Space by Unit Cubes.
Mutations in finite groups.
Non-integer lattice tilings by (k, n) truncated crosses.
Number of solutions in a box of a linear equation in an Abelian group
For every finite Abelian group Γ and for all , if there exists a solution of the equation in non-negative integers , where are positive integers, then the number of such solutions is estimated from below in the best possible way.
Number of solutions in a box of a linear homogeneous equation in an Abelian group
On A Combinatorial Problem Connected withFactorizations
On a complete set of operations for factorizing codes
It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set of operations exists such that each factorizing code can be obtained by using the operations in and starting with prefix or suffix codes. is named here a complete set of operations (for factorizing codes). We show...
On a complete set of operations for factorizing codes
It is known that the class of factorizing codes, i.e., codes satisfying the factorization conjecture formulated by Schützenberger, is closed under two operations: the classical composition of codes and substitution of codes. A natural question which arises is whether a finite set O of operations exists such that each factorizing code can be obtained by using the operations in O and starting with prefix or suffix codes. O is named here a complete set of operations (for factorizing codes). We show...
On a conjecture of Erdös and Heilbronn
On a conjecture of Hamidoune for subsequence sums.
On a connection of number theory with graph theory
We assign to each positive integer a digraph whose set of vertices is and for which there is a directed edge from to if . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.