No description available.

No description available.

No description available.

This dissertation demonstrates the existence of a pair of algebraic and geometric structures on triangles of groups and on quadrilaterals of groups. These structures are an automatic and biautomatic structure. In addition, this paper also discusses the growth function for the quadrilaterals. We show that these groups have these desired structures and discuss what they are. We also give an extraordinary example of a pair of quadrilaterals of groups that are defined nearly identically but do not behave alike. / Ph. D.

Let S be a semigroup with zero and let a S\O. Denote by V(a) the set of all inverses of a, that is, V(a) = (x ∈ S: axa=a. xax=x). Let n be a fixed positive integer. A semigroup S with zero is said to be homogeneous n regular if the cardinal number of the set V(a) of all inverses of a is n for every nonzero element a in S. Let T be a subset of S. We denote by E(T) the set of all idempotents of S in T. The next theorem is a generalization of R. McFadden and Hans Schneider's theorem [1] . Theorem 1. Let S be a 0-simple semigroup and let n be a fixed positive integer. Then the following are equivalent. (i) S is a homogeneous n regular and completely 0-simple semigroup. (ii) For every a≠0 in S there exist precisely n distinct nonzero elements (xᵢ)_{i [= symbol with an n on top]l} such that axᵢa=a for i=1, 2, ..., n and for all c, d in S cdc=c≠0 implies dcd=d. (iii) For every a≠0 in S there exist precisely h distinct nonzero idempotents (eᵢ)_{i [= symbol with an h above]l} Eₐ and k distinct nonzero idempotents (fⱼ)_{j[= symbol with a k above]}= Fₐ such that eᵢa=a=afⱼ for i =1, 2, …, h, j = 1, 2, …, k hk=n, Eₐ contains every nonzero idempotent which is a left unit of a, Fₐ contains every nonzero idempotent which is a right unit of a and Eₐ ⋂ Fₐ contains at most one element. (iv) For every a≠0 in S there exist precisely k nonzero principal right ideals (Rᵢ)_{i[= symbol with a k above]1} and h nonzero principal left ideals (Lⱼ)_{j[= symbol with h above]1} such that Rᵢ and Lⱼ contain h and k inverses of a, respectively, every inverse of a is contained in a suitable set Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h and Rᵢ ⋂ Lⱼ for i = 1, 2, .., k, j = 1, 2, .., h, and Rᵢ ⋂ Lⱼ contains at most one nonzero idempotent, where hk = n. (v) Every nonzero principal right ideal R contains precisely h nonzero idempotents and every nonzero principal left ideal L contains precisely k nonzero idempotents such that hk=n, and R⋂L contains at most one nonzero idempotent. (vi) S is completely 0-simple. For every 0-minimal right ideal R there exist precisely h 0-minimal left ideals (Li)_{i[= symbol with an h above]1} and for every 0-minimal left ideal L there exist precisely k 0-minimal right ideals (Rj)_{j[= symbol with a k above]1} such that LRⱼ=LiR=S, for every i=1,2,..,h, j=l,2,.. ,k, where hk=n. (vii) S is completely 0-simple. Every 0-minimal right ideal R of S is the union of a right group with zero G°, a union of h disjoint groups except zero, and a zero subsemigroup Z uhich annihilates the right ideal R on the left and every 0-minimal left ideal L of S is the union of a left group with zero G’° a union of k disjoint groups except zero, and a zero subsemigroup Z' which annihilates the left ideal L on the right and hk=n. (viii) S contains at least n nonzero distinct idempotents, and for every nonzero idempotent e there exists a set E of n distinct nonzero idempotents of S such that eE is a right zero subsemigroup of S containing precisely h nonzero idempotents, Ee is a left zero subsemigroup of S containing precisely k nonzero idempotents of S, e (E(S)\E) = (0) = (E(S)\E)e, and eE⋂Ee = (e), where hk=n. S is said to be h-k type if every nonzero principal left ideal of S contains precisely k nonzero idempotents and every nonzero principal right ideal of S contains precisely h nonzero idempotents of S. W. D. Munn defined the Brandt congruence [2]. A congruence ρ on a sernigroup S with zero is called a Brandt congruence if S/ρ is a Brandt semigroup. Theorem 2. Let S be a 1-n type homogeneous n regular and complete:y 0-simple semigroup. Define a relation ρ on S in such a way that a ρb if and only if there exists a set (eᵢ) _{i[=symbol with an n above]1} of n distinct nonzero idempotents such that eᵢa=ebᵢ≠0, for every i=1, 2, . , n. Then ρ is an equivalence S\0. If we extend ρ on S by defining (0) to be ρ-class on S, then ρ is a proper Brandt congruence on S, then ρ ⊂ σ. Let P=(pᵢⱼ) be any n x n matrix over a group with G°, and consider any n distinct points A₁, A₂, . , A_n in the plane, which we shall call vertices. For every nonzero entry pᵢⱼ≠0 of the matrix P, we connect the vertex Aᵢ to the vertex Aⱼ by means of a path [a bar over both AᵢAⱼ] which we shall call an edge (a loop if i = j) directed from Aᵢ to Aⱼ. In this way, with every n x n matrix P can be associated a finite directed graph G(P). Let S=M°(G;In,In;P) be a Rees matrix semigroup. Then the graph G(P) is called the associated graph of the semigroup S, or simply it is the graph G(P) of S. Theorem 3. A Rees matrix semigroup S=M°(G;In,In;P) is homogenous m² regular if the directed graph G(P) of the semigroup S is regular of degree m [3, p. 11]. / Doctor of Philosophy

The formula and theory for estimating a missing value in the case of covariance in a randomized block has been presented in this paper. It has also been found that the formula given corresponds to Yates’ formula for a missing value in a randomized block when there is only one variable present in the experiment. / Master of Science

<p dir="ltr">A group is said to be matricially (Frobenius) stable if every function from the group to unitary matrices that is "almost multiplicative" in the point operator (Frobenius) norm topology is "close" to a genuine unitary representation in the same topology. A result of Dadarlat shows that for a large class of groups, non-torsion even cohomology obstructs matricial stability. However, the proof doesn't generate explicit almost multiplicative maps that are far from genuine representations. In this paper, we compute explicit almost homomorphisms for all finitely generated groups with a non-torsion 2-cohomology class with a residually finite central extension. We use similar techniques to show that finitely generated nilpotent groups are Frobenius stable if and only if they are virtually cyclic, and that a finitely generated group with a non-torsion 2-cohomology class that can be written as a cup product of two 1-cohomology classes is not Frobenius stable.</p><p><br></p>

This work explores the unsolvability of the general quintic equation through the lens of Galois theory. We begin by providing a historical perspective on the problem. This starts with the solution of the general cubic equation derived by Italian mathematicians. We then move on to Lagrange's insights on the importance of studying the permutations of roots. Finally, we discuss the critical contributions of Évariste Galois, who connected the solvability of polynomials to the properties of permutation groups. Central to our thesis is the introduction and motivation of key concepts such as fields, solvable groups, Galois groups, Galois extensions, and radical extensions. We rigorously develop the theory that connects the solvability of a polynomial to the solvability of its Galois group. After developing this theoretical framework, we go on to show that there exist quintic polynomials with Galois groups that are isomorphic to the symmetric group S5. Given that S5 is not a solvable group, we establish that the general quintic polynomial is not solvable by radicals. Our work aims to provide a comprehensive and intuitive understanding of the deep connections between polynomial equations and abstract algebra.

The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.