scholarly journals LOGARITHMIC SIGNATURES FOR ABELIAN GROUPS AND THEIR FACTORIZATION

2013 ◽  
Vol 57 (1) ◽  
pp. 21-33 ◽  
Author(s):  
Pavol Svaba ◽  
Tran van Trung ◽  
Paul Wolf
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Efficient Algorithms ◽  
Finite Abelian Groups ◽  
Logarithmic Signatures ◽  
Vital Importance ◽  
Essential Component ◽  
Factorization Algorithms ◽  
Factoring Algorithm

ABSTRACT Factorizable logarithmic signatures for finite groups are the essential component of the cryptosystems MST1 and MST3. The problem of finding efficient algorithms for factoring group elements with respect to a given class of logarithmic signatures is therefore of vital importance in the investigation of these cryptosystems. In this paper we are concerned about the factorization algorithms with respect to transversal and fused transversal logarithmic signatures for finite abelian groups. More precisely we present algorithms and their complexity for factoring group elements with respect to these classes of logarithmic signatures. In particular, we show a factoring algorithm with respect to the class of fused transversal logarithmic signatures and also its complexity based on an idea of Blackburn, Cid and Mullan for finite abelian groups.

scholarly journals Pinned-flags of some operations on fuzzy subgroups

2005 ◽  
Vol 2005 (23) ◽  
pp. 3819-3826
Author(s):  
B. B. Makamba ◽  
V. Murali
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Fuzzy Subgroups

Fuzzy subgroups of finite groups have been treated recently using the concept of pinned-flags. In this paper, we consider the operations of intersection, sum, product, and quotient of fuzzy subgroups of finite abelian groups in general, in terms of pinned-flags. We develop algorithms to construct pinned-flags of fuzzy subgroups corresponding to these operations and prove their validity. We illustrate some applications of such algorithms.

scholarly journals COMMUTATIVITY DEGREE OF A CLASS OF FINITE GROUPS AND CONSEQUENCES

2013 ◽  
Vol 88 (3) ◽  
pp. 448-452 ◽  
Author(s):  
RAJAT KANTI NATH
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Finite Abelian Groups ◽  
A Finite Group ◽  
Open Question

AbstractThe commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The object of this paper is to compute the commutativity degree of a class of finite groups obtained by semidirect product of two finite abelian groups. As a byproduct of our result, we provide an affirmative answer to an open question posed by Lescot.

scholarly journals Asymptotic estimates on finite Abelian groups

2003 ◽  
Vol 74 (88) ◽  
pp. 57-70
Author(s):  
C. Calderon
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Counting Function ◽  
Divisor Problem ◽  
Arithmetic Functions ◽  
Asymptotic Estimates ◽  
Finite Abelian Groups

By using Ivic's methods for general divisor problem and counting function of abelian finite groups, we obtain results related to several arithmetic functions.

scholarly journals EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

On the probability that an automorphism of a group fixes an element of the group

2019 ◽  
Vol 19 (10) ◽  
pp. 2050198
Author(s):  
Ashish Goyal ◽  
Hemant Kalra ◽  
Deepak Gumber
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
A Finite Group

Let [Formula: see text] be a finite group and let [Formula: see text] denote the probability that a randomly chosen element from [Formula: see text] fixes a randomly chosen element from [Formula: see text]. We classify all finite abelian groups [Formula: see text] such that [Formula: see text] in the cases when [Formula: see text] is the smallest prime dividing [Formula: see text], and when [Formula: see text] is any prime. We also compute [Formula: see text] for some classes of finite groups. As a consequence of our results, we deduce that if [Formula: see text] is a finite [Formula: see text]-group having a cyclic maximal subgroup, then [Formula: see text] divides [Formula: see text].

On the Isomorphism of Integral Group Rings. I

1969 ◽  
Vol 21 ◽  
pp. 410-413 ◽  
Author(s):  
Sudarshan K. Sehgal
Keyword(s):  
Finite Groups ◽  
Classical Result ◽  
Abelian Groups ◽  
Group Rings ◽  
Integral Group ◽  
Normal Subgroups ◽  
Finite Abelian Groups ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

In this note we study the question of automorphisms of the integral group ring Z(G) of a finite group G. We prove that if G is nilpotent of class two, any automorphism of Z(G) is composed of an automorphism of G and an inner automorphism by a suitable unit of Q(G), the group algebra of G with rational coefficients. In § 3, we prove that if two finitely generated abelian groups have isomorphic integral group rings, then the groups are isomorphic. This is an extension of the classical result of Higman (2) for the case of finite abelian groups. In the last section we give a new proof of the fact that an isomorphism of integral group rings of finite groups preserves the lattice of normal subgroups. Other proofs are given in (1;4).

scholarly journals EFFICIENT ALGORITHMS FOR THE BASIS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 03 (04) ◽  
pp. 537-552 ◽  
Author(s):  
GREGORY KARAGIORGOS ◽  
DIMITRIOS POULAKIS
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Deterministic Algorithm ◽  
Efficient Algorithms ◽  
Finite Abelian Groups ◽  
Prime Factorization ◽  
Deterministic Algorithms ◽  
Generating System

In this paper we consider the problem of computation of a basis for an abelian group G with N elements such that the prime factorization of N is known. We present two deterministic algorithms for this task and a deterministic algorithm in case where a generating system for G is given.

DESCRIPTIVE COMPLEXITY OF FINITE ABELIAN GROUPS

2010 ◽  
Vol 20 (08) ◽  
pp. 1087-1116 ◽  
Author(s):  
WALID GOMAA
Keyword(s):  
Positive Integer ◽  
Finite Groups ◽  
Abelian Groups ◽  
Upper And Lower Bounds ◽  
Single Element ◽  
Finite Abelian Groups ◽  
Dihedral Groups ◽  
Unpublished Paper ◽  
Descriptive Complexity ◽  
First Order

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht–Fraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω( log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by Koponen and Luosto in an unpublished paper).

scholarly journals Strongly aperiodic logarithmic signatures

2013 ◽  
Vol 7 (2) ◽  
pp. 147-179
Author(s):  
Reiner Staszewski ◽  
Tran van Trung
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Public Key ◽  
Logarithmic Signatures ◽  
Public Key Cryptosystems ◽  
Private Key ◽  
Main Component

Abstract. Logarithmic signatures for finite groups are the essential constituent of public key cryptosystems and . Especially they form the main component of the private key of . Regarding the use of , it has become a vital issue to construct new classes of logarithmic signatures having features that do not share with the well-known class of transversal or fused transversal logarithmic signatures. For this purpose Baumeister and de Wiljes recently presented an interesting method of constructing aperiodic logarithmic signatures for abelian groups. In this paper we introduce the concept of strongly aperiodic logarithmic signatures and show their constructions for abelian p-groups on the basis of the Baumeister–de Wiljes method.

scholarly journals Integral Sets and Cayley Graphs of Finite Groups

10.37236/2053 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Roger C. Alperin ◽  
Brian L. Peterson
Keyword(s):  
Boolean Algebra ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Finite Abelian Groups

Integral sets of finite groups are discussed and related to the integral Cayley graphs. The Boolean algebra of integral sets are determined for dihedral group and finite abelian groups. We characterize the finite abelian groups as those finite groups where the Boolean algebra generated by  integral sets equals  the Boolean algebra generated by its subgroups.

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