This thesis is a study of primary Abelian groups and height.

"In this paper, we will use additive notation and will let O be the identity element of our groups. Also, let it be agreed that by "group" we mean "abelian group." First, we wish to consider cyclic groups. A group G is said to be cyclic if it can be generated by a single element, i.e., there is an element a in G such that all other elements in G are integral multiples of a. If G is infinite, it is isomorphic to the additive group opf integers. If G has n elements, G is isomorphic to the additive group of integers mod n"--Chapter 1. / Typescript. / "June, 1959." / "Submitted to the Graduate School of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references.

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Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / In this paper we have determined the structure of divisible groups, certain primary groups, and countable torsion groups. Chapter 1 introduces two important infinite abelian groups, R and Z(p^∞). The structure of these groups is completely known and we have given most of the important properties of these groups in Chapter 1. Of special importance is the fact that a divisible group can be decomposed into a direct sum of groups each isomorphic to R or Z(p^∞). This is Theorem 2.12 and it classifies all divisible groups in terms of these two well-known groups. Theorem 1.6 is of great importance since it reduces the study of torsion groups to that of primary groups. We now have that Theorems 3.3 and 5.5 apply to countable torsion groups as well as primary groups. Theorem 3.3 gives a necessary and sufficient condition for an infinite torsion group to be a direct sum of cyclic groups. These conditions are more complicated than the finite case. From Theorem 3.3, we derived Corollary 3.5. This result is used later on to get that the Ulm factors of a group are direct sums of cyclic groups. In essence, Ulm's theorem says that a countable reduced primary group can be determined by knowing its Ulm type and its Ulm sequence. Now by Corollary 3.5, we have only to look at the number of cyclic direct summands of order p^n (for all integers n) for each Ulm factor. This gives us a system of invariants which we can assign to the group. Once again, these invariants are much harder to arrive at than in the finite case. / 2999-01-01

Let K/F be a separable extension. (i) If K = F(α) with αⁿ ∈ F for some n, K/F is said to be a radical extension. (ii) If there exists a sequence of fields F = F₀ ⊆ F₁ ⊆ ... ⊆ F(s) = K so that Fᵢ₊₁ = Fᵢ(αᵢ) with αᵢⁿ⁽ⁱ⁾ ∈ Fᵢ for some nᵢ ∈ N, charF ∧nᵢ for every i, and [Fᵢ₊₁ : Fᵢ] = nᵢ, K/F is said to be a radical tower. In the first part of this work, we present two theorems which give sufficient conditions for a field extension K/F to be radical. In the second part, we present results which provide conditions under which every subfield of a radical tower is also a radical tower.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 79-81. / by Ranee Kathryn Gupta. / Ph.D.

The purpose of this paper was to find a general formula to count the number of automorphisms of any finite abelian group. These groups were separated into five different types. For each of the first three types, theorems were proven, and formulas were derived based on the theorems. A formula for the last two types of groups was derived from a theorem based on a conjecture which was proven in only one direction. Then it was shown that a count found from any of the first three formulas could also be found using the last formula. The result of these comparisons gave credence to the conjecture. Thus we found that the last formula is a general formula to count the number of automorphisms of finite abelian groups. / Department of Mathematical Sciences

Ribosomal RNA genes (rDNA) exist in yeast both as a single chromosomal array of tandemly repeated units and as extrachromosomal units named 3um plasmids, although the relationship between these two forms is unclear. Inheritance of rDNA was studied using two systems. The first used a naturally occuring rDNA restriction enzyme polymorphism between two strains to distinguish between their rDNA arrays, and the second involved cloning a tRNA suppressor gene into rDNA to label individual rDNA units. An added interest to the study of the inheritance of rDNA in yeast was the possible association between it and the inheritance of the Psi factor, an enigmatic type of nonsense suppressor in yeast which shows extra-chromosomal inheritance. In a cross heterozygous for the rDNA polymorphism and the psi factor, tetrad analysis suggested that the psi factor had segregated 4:0. The majority of the rDNA units segregated in a 2:2 fashion, which suggested that reciprocal recombination in the rDNA of psi⁺ diploids is heavily suppressed as was previously shown for psi^- diploids. A heterologous plasmid containing the tRNA suppressor gene was constructed and transformed into haploid and diploid hosts. A series of transformants was obtained and physical and genetic analysis suggested that they contained tRNA suppressor gene(s) integrated into their rDNA. In a cross heterozygous for rDNA-tRNA gene insert(s), 6% of the tetrads dissected showed a meiotic segregation of the suppressed phenotype which could most probably be accounted for by inter-chromosomal gene conversion. This observation could be interpreted in two ways. Firstly, recombination intermediates between rDNA on homologues may occur in meiosis, but they are mostly resolved as gene conversions without reciprocal cross-over. Alternatively, gene conversion tracts in rDNA are rare but very long so that the tRNA gene insert was always included in the event. 3um rDNA plasmids containing the tRNA gene marker were not detected in any of the transformants analysed. An extensive quantitative analysis of the rate of reversion of the suppressed phenotype amongst these transformants identified a particulary unstable transformant group. It was proposed that the mechanism of reversion was loss of the tRNA gene insert by unequal sisterstrand exchange, and the mechanism was shown to be independent of the recombination/repair genes RAD1, RAD52, and RAD51. A genetic analysis of stability suggested that there may have been at least two loci segregating in the host strains with additive effects on stability.

"Definition 1. A group G is an extension of a group A by a group B if and only if A is a normal subgroup of G and the factor group G/A is isomorphic to B. Definition 2. Two extensions G and H of A by B are called equivalent if and only if there exists an isomorphism between G and H that on A coincides with the identity automorphism and that maps onto each other the cosets of A corresponding to one and the same element of B. Consider the following example: let G be the cyclic group of order 4, that is G = {1, a, a², a³} and let H[subscript G] = {1, a²} be a normal subgroup of G. Now let V be the Klein four-group, that is, V = {1, a, b, c : a²=b²=c²=1} and H[subscript V] = {1, b} a normal subgroup of V. Since H[subscript G] and H[subscript V] are cyclic groups of order 2, set H[subscript V] = H[subscript G] = H. G and V are extensions of H by itself but are not equivalent extensions since no isomorphism exists between G and V. So the question arises: what are the necessary and sufficient conditions that two extensions G and G' of a group A by a group B be equivalent?"--Introduction. / Typescript. / "January 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaf 37).