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