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 Title
 Finite valuated groups.
 Creator
 Holroyd, Keiko Ito, Florida Atlantic University, Richman, Fred
 Abstract/Description

The concept of valuated groups, simply presented groups, and simultaneous decomposition of an abelian group and a subgroup are discussed. We classify the structure of finite valuated pgroups of order up p^4. With a refinement of a classical theorem on bounded pure subgroups, we also relate the decomposition of a finite valuated pgroup to the simultaneous decomposition of a finite abelian pgroup and a subgroup.
 Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/15320
 Subject Headings
 Finite groups, Abelian groups, Abelian pgroups, Algebra, Homological
 Format
 Document (PDF)
 Title
 Defeating pattack in nonabelian discrete logarithm problem.
 Creator
 Magar, Krishna Thapa, Ilic, Ivana, Magliveras, Spyros S., Graduate College
 Date Issued
 20130412
 PURL
 http://purl.flvc.org/fcla/dt/3361325
 Subject Headings
 NonAbelian groups, Logarithms
 Format
 Document (PDF)
 Title
 A Constructive Theory of Ordered Sets and their Completions.
 Creator
 Joseph, Jean S., Richman, Fred, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The context for the development of this work is constructive mathematics without the axiom of countable choice. By constructive mathematics, we mean mathematics done without the law of excluded middle. Our original goal was to give a list of axioms for the real numbers R by only considering the order on R. We instead develop a theory of ordered sets and their completions and a theory of ordered abelian groups.
 Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013007
 Subject Headings
 Constructive mathematics, Ordered sets, Abelian groups
 Format
 Document (PDF)
 Title
 A computation of the Hall coefficient g(q)[('6,4,2)(,42)(,4,2)].
 Creator
 Anez, Myriam T., Florida Atlantic University, Schmidmeier, Markus
 Abstract/Description

Let L be a uniserial ring of length n, with maximal ideal r , and finite residue field Λ/ r . We consider Λmodules which possess a finite composition series. We note that a Λmodule has the form B ≅ ⨁i=1m Λ/ rli , where the type of B is the partition l = ( l1,&ldots;,lm ) denoted by t(B). For Λmodules A, B, C with t(A) = m , t(B) = l , t(C) = n , if A ⊆ B, and B/A ≅ C, we define GBAC = {U ⊆ B : U ≅ A and B/U ≅ C}. We show that GBAC = MonoA,B,C Aut A =  S (A, B, C)/∼ = glmn (q),...
Show moreLet L be a uniserial ring of length n, with maximal ideal r , and finite residue field Λ/ r . We consider Λmodules which possess a finite composition series. We note that a Λmodule has the form B ≅ ⨁i=1m Λ/ rli , where the type of B is the partition l = ( l1,&ldots;,lm ) denoted by t(B). For Λmodules A, B, C with t(A) = m , t(B) = l , t(C) = n , if A ⊆ B, and B/A ≅ C, we define GBAC = {U ⊆ B : U ≅ A and B/U ≅ C}. We show that GBAC = MonoA,B,C Aut A =  S (A, B, C)/∼ = glmn (q), where Λ/ r  = q, and the last equality comes from evaluating the Hall polynomial glmn (t) ∈ Z [t] at q, as stated in Hall's Theorem. We note that GBAC make up the coefficients of the Hall algebra. We provide a proof that the Hall algebra is a commutative and associative ring. Using the property of associativity of the Hall algebra and I. G. MacDonald's formula: glb1l =qnl nbn 1li≥ 1l'i b'i,b' il'i+1 q1 we develop a procedure to generate arbitrary Hall polynomials and we compute g6,4,2 4,24,2 (q).
Show less  Date Issued
 2005
 PURL
 http://purl.flvc.org/fcla/dt/13289
 Subject Headings
 Mathematical statistics, Algebra, Abstract, Abelian groups
 Format
 Document (PDF)
 Title
 The existence of minimal logarithmic signatures for classical groups.
 Creator
 Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
Show less  Date Issued
 2011
 PURL
 http://purl.flvc.org/FAU/3172943
 Subject Headings
 Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
 Format
 Document (PDF)
 Title
 On the minimal logarithmic signature conjecture.
 Creator
 Singhi, Nidhi., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups,...
Show moreThe minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BNpair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS.
Show less  Date Issued
 2011
 PURL
 http://purl.flvc.org/FAU/3172946
 Subject Headings
 Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
 Format
 Document (PDF)
 Title
 Subgroups of bounded Abelian groups.
 Creator
 Petroro, Carla., Florida Atlantic University, Schmidmeier, Markus
 Abstract/Description

Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pnbounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) > (A', B') are morphisms f : B > B' such that f (A) > A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the KrullRemakSchmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities...
Show moreBirkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pnbounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) > (A', B') are morphisms f : B > B' such that f (A) > A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the KrullRemakSchmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.
Show less  Date Issued
 2004
 PURL
 http://purl.flvc.org/fcla/dt/13118
 Subject Headings
 Abelian groups, Modules (Algebra), Indecomposable modules, Representations of groups, Algebras, Linear
 Format
 Document (PDF)
 Title
 Avoiding abelian squares in infinite partial words.
 Creator
 Severa, William., Harriet L. Wilkes Honors College
 Abstract/Description

Famous mathematician Paul Erdèos conjectured the existence of infinite sequences of symbols where no two adjacent subsequences are permutations of one another. It can easily be checked that no such sequence can be constructed using only three symbols, but as few as four symbols are sufficient. Here, we expand this concept to include sequences that may contain 'do not know'' characters, called holes. These holes make the undesired subsequences more common. We explore both finite and infinite...
Show moreFamous mathematician Paul Erdèos conjectured the existence of infinite sequences of symbols where no two adjacent subsequences are permutations of one another. It can easily be checked that no such sequence can be constructed using only three symbols, but as few as four symbols are sufficient. Here, we expand this concept to include sequences that may contain 'do not know'' characters, called holes. These holes make the undesired subsequences more common. We explore both finite and infinite sequences. For infinite sequences, we use iterating morphisms to construct the nonrepetitive sequences with either a finite number of holes or infinitely many holes. We also discuss the problem of using the minimum number of different symbols.
Show less  Date Issued
 2010
 PURL
 http://purl.flvc.org/FAU/3335460
 Subject Headings
 Abelian groups, Mathematics, Study and teaching (Higher), Combinatorial analysis, Combinatorial set theory, Probabilities
 Format
 Document (PDF)