This thesis deals with various problems about the normal and subnormal structure of infinite groups. We first consider the relationship between the number of normal subgroups of a group G and of a subgroup H of finite index in G. We prove Theorem 1.5 There exists a finitely generated group G which has a subgroup H of index 2 such that H has continuously many normal subgroups and G has only countably many normal subgroups. Proposition 1.7 Let k be an infinite cardinal. Then there exists a group G of cardinality k that has only 12 normal subgroups but which contains a subgroup H of index 2 having k normal subgroups. We then consider partially ordered sets and investigate the subnormal structure of generalized wreath products. We deal with the question whether the number of subnormal subgroups of an infinite group is determined by the number of its n-step subnormal subgroups for an integer n. We prove Theorem 5.3 Let G be a group. Then G has finitely many subnormal subgroups if and only if it has finitely many 2-step subnormal subgroups. Theorem 5.5 Let m and n be infinite cardinals such that m ≤ n. Then there exists a group G with the following properties: (1) The cardinality of G is n. (2) The number of normal subgroups of G is <mathematical symbol>. (3) The number of 2-step subnormal subgroups of G is m. (4) The number of 3-step subnormal subgroups of G is 2ⁿ. Finally we consider characteristically simple groups with countably many normal subgroups. We construct a new type of characteristically simple groups: Corollary 6.15 Let ∧ be a partially ordered set such that for λ,<mathematical symbol>∊∧ there exists an automorphism a of ∧ such that <mathematical symbol> ≤ λa. Let <mathematical symbol>(∧) be the distributive lattice of semi-ideals of ∧. Then there exists a group G with the following properties: (1) |G| ≤ max(<mathematical symbol> of |<mathematical symbol>(∧)|). (2) All subnormal subgroups of G are normal in G. (3) The lattice of normal subgroups of G is isomorphic to <mathematical symbol> (∧). (4) The group G is characteristically simple.

In this thesis a set, Ω, of cardinality N_K and a group acting on Ω, with N_K+1 orbits on the power set of Ω, is found for every infinite cardinal N_K. Let W_K denote the initial ordinal of cardinality N_K. Define N := {α₁α₂ . . . α_n∣ 0 < n < w, α_j ∈ w_K for j = 1, . . .,n, α_n a successor ordinal} R := {ϰ ∈ N ∣ length(ϰ) = 1 mod 2} and let these sets be ordered lexicographically. The order types of N and R are Κ-types (countable unions of scattered types) which have cardinality N_K and do not embed w*₁. Each interval in N or R embeds every ordinal of cardinality N_K and every countable converse ordinal. N and R then embed every K-type of cardinality N_K with no uncountable descending chains. Hence any such order type can be written as a countable union of wellordered types, each of order type smaller than w^w_k. In particular, if α is an ordinal between w^w_k and w_K+1, and A is a set of order type α then A= ⋃_n<wA_n where each A_n has order type wⁿ_k. If X is a subset of N with X and N - X dense in N, then X is orderisomorphic to R, whence any dense subset of R has the same order type as R. If Y is any subset of R then R is (finitely) piece- wise order-preserving isomorphic (PWOP) to R ⋃^. Y. Thus there is only one PWOP equivalence class of N_K-dense K-types which have cardinality N_K, and which do not embed w*₁. There are N_K+1 PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N_K+1 orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N₀ orbits on R if ∣k∣ is smaller than w₁ and ∣k∣ if it is not smaller.

Thesis (M.S.)--State University of New York at Binghamton, Computer Science Department, 2006. / Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 51-52).

The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$ and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and $G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of $F$ and $T$ were characterised in the seminal paper of Brin ([16]) and, later on, Brin and Guzman ([17]) investigate automorphisms of $T_{n, n-1}$ and $F_{n}$ for $n > 2$. However, their techniques give no information about automorphisms of $G_{n,r}$. The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article [10], we show that automorphisms of $G_{n,r}$ are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition. In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article [6] of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on [44]. We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure. Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h−1 f (h)\phi$. This defines an equivalence relation on $G$ and $G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$. In the final chapter we show, using the description of $\aut{G_{n,r}}$, that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta dissertaÃÃo apresenta resultados sobre coberturas de grupos por sub-grupos abelianos, subgrupos de Sylow e subgrupos normais. O Teorema de Neumann à indispensÃvel no estudo das coberturas por subgrupos. Apresentamos no apÃndice C uma prova elementar de um resultado muito importante nas coberturas p-Sylow. / The paper results on the Coverage groups by abelian subgroups, subgroups of Sylow and normal subgroups. We present in appendix C an elementary proof a very important result in the coverage p-Sylow.

This thesis is concerned with aspects of the model theory of infinite soluble groups. The results proved lie on the border between group theory and model theory: the questions asked are of a model-theoretic nature but the techniques used are mainly group-theoretic in character. We present a characterization of those groups contained in the universal closure of a restricted wreath product U wr G, where U is an abelian group of zero or finite square-free exponent and G is a torsion-free soluble group with a bound on the class of its nilpotent subgroups. For certain choices of G we are able to use this characterization to prove further results about these groups; in particular, results related to the decidability of their universal theories. The latter part of this work consists of a number of independent but related topics. We show that if G is a finitely generated abelian-by-metanilpotent group and H is elementarily equivalent to G then the subgroups gamma_n(G) and gamma_n(H) are elementarily equivalent, as are the quotient groups G/gamma_n(G) and G/gamma_n(H). We go on to consider those groups universally equivalent to F_2(VN_c), where the free groups of the variety V are residually finite p-groups for infinitely many primes p, distinguishing between the cases when c = 1 and when c > 2. Finally, we address some important questions concerning the theories of free groups in product varieties V_k · · ·V_1, where V_i is a nilpotent variety whose free groups are torsion-free; in particular we address questions about the decidability of the elementary and universal theories of such groups. Results mentioned in both of the previous two paragraphs have applications here.

We broaden the theory of dynamical interpretation, investigate the property of commutativity and explore the subject of subgroups forming free products in Thompson's group V. We expand Brin's terminology for a revealing pair to an any tree pair. We use it to analyse the dynamical behaviour of an arbitrary tree pair which cannot occur in a revealing pair. Hence, we design a series of algorithms generating Brin's revealing pair from any tree pair, by successively eliminating the undesirable structures. To detect patterns and transitioning between tree pairs, we introduce a new combinatorial object called the chains graph. A newly defined, unique and symmetrical type of a tree pair, called a balanced tree pair, stems from the use of the chains graphs. The main theorem of Bleak et al. in "Centralizers in the R. Thompson's Group V_n" states the necessary structure of the centraliser of an element of V. We provide a converse to this theorem, by proving that each of the predicted structures is realisable. Hence we obtain a complete classification of centralisers in V. We give an explicit construction of an element of V with prescribed centraliser. The underlying concept is to embed a Cayley graph of a finite group into the flow graph (introduced in Bleak et al.) of the desired element. To reflect the symmetry, we present the resulting element in terms of a balanced tree pair. The group V is conjectured to be a universal coCF group, which generates interest in studying its subgroups. We develop a better understanding of embeddings into V by providing a necessary and sufficient dynamical condition for two subgroups (not both torsion) to form a free product in V. For this, we use the properties, explored in Bleak and Salazar-Díaz "Free Products in Thompson's Group V", of sets of so--called important points, and the Ping-Pong action induced on them.