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Finite valuated groups
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 p-groups 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 p-group to the simultaneous decomposition of a finite abelian p-group and a subgroup.
Date Issued
1996
PURL
http://purl.flvc.org/fcla/dt/15320
Subject Headings
Finite groups, Abelian groups, Abelian p-groups, Algebra, Homological
Format
Document (PDF)
Defeating p-attack in non-abelian discrete logarithm problem
Title
Defeating p-attack in non-abelian discrete logarithm problem.
Creator
Magar, Krishna Thapa, Ilic, Ivana, Magliveras, Spyros S., Graduate College
Date Issued
2013-04-12
PURL
http://purl.flvc.org/fcla/dt/3361325
Subject Headings
Non-Abelian groups, Logarithms
Format
Document (PDF)
A Constructive Theory of Ordered Sets and their Completions
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)
computation of the Hall coefficient g(q)[('6,4,2)(,42)(,4,2)]
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 Λ-modules which possess a finite composition series. We note that a Λ-module 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 Λ-modules 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 Λ-modules which possess a finite composition series. We note that a Λ-module 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 Λ-modules 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 -nb-n 1li≥ 1l'i -b'i,b' i-l'i+1 q-1 we develop a procedure to generate arbitrary Hall polynomials and we compute g6,4,2 4,24,2 (q).
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Date Issued
2005
PURL
http://purl.flvc.org/fcla/dt/13289
Subject Headings
Mathematical statistics, Algebra, Abstract, Abelian groups
Format
Document (PDF)
existence of minimal logarithmic signatures for classical groups
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.
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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)
the minimal logarithmic signature conjecture
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 BN-pair, 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.
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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)
Subgroups of bounded Abelian groups
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 pn-bounded 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 Krull-Remak-Schmidt 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 pn-bounded 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 Krull-Remak-Schmidt 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.
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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)
Avoiding abelian squares in infinite partial words
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 non-repetitive 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.
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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)