The lattice of normal subgroups of an infinite group

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 . (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,∊∧ there exists an automorphism a of ∧ such that ≤ λa. Let (∧) be the distributive lattice of semi-ideals of ∧. Then there exists a group G with the following properties: (1) |G| ≤ max( of |(∧)|). (2) All subnormal subgroups of G are normal in G. (3) The lattice of normal subgroups of G is isomorphic to (∧). (4) The group G is characteristically simple.