Book contents
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Groups with all non-subnormal subgroups of finite rank
Published online by Cambridge University Press: 15 December 2009
- Frontmatter
- Contents
- Introduction
- Gracefulness, group sequencings and graph factorizations
- Orbits in finite group actions
- Groups with finitely generated integral homologies
- Invariants of discrete groups, Lie algebras and pro-p groups
- Groups with all non-subnormal subgroups of finite rank
- On some infinite dimensional linear groups
- Groups and semisymmetric graphs
- On the covers of finite groups
- Groupland
- On maximal nilpotent π-subgroups
- Characters of p-groups and Sylow p-subgroups
- On the relation between group theory and loop theory
- Groups and lattices
- Finite generalized tetrahedron groups with a cubic relator
- Character degrees of the Sylow p-subgroups of classical groups
- Character correspondences and perfect isometries
- The characters of finite projective symplectic group PSp(4, q)
- Exponent of finite groups with automorphisms
- Classifying irreducible representations in characteristic zero
- Lie methods in group theory
- Chevalley groups of type G2 as automorphism groups of loops
Summary
Abstract
In the present paper we start to study the soluble groups in which every non-subnormal subgroup has finite special rank.
Introduction
Let G be a group, Lnon–sn(G) the set of all non-subnormal subgroups of G. If Lnon–sn(G) = Ø then we obtain a group, every subgroup of which is subnormal. The study of these groups was very fruitful and has brought many interesting results (see the books [12], [13], [8]). The groups G in which the set Lnon–sn(G) is “very small” in some sense is the natural next consideration. For many domains of Infinite Group Theory “to be very small” means to satisfy some finiteness conditions. The first natural finiteness conditions in Group Theory were the classical minimal and maximal conditions. The groups with minimal condition for non-subnormal subgroups have been considered by S. Franciosi and F. de Giovanni [1]. The groups with dual maximal condition for non-subnormal subgroups have been studied by L.A. Kurdachenko and H. Smith [4]. The groups with maximal condition for non-subnormal subgroups is nearly allied to the groups, in which the set Lnon–sn(G) consists only of the finitely generated subgroups. Such groups have been considered. The results of papers [1], [4] have been extended on groups with weak minimal and maximal condition on non-subnormal subgroups [5], [6]. The weak minimal and maximal conditions are connected with the concept of special rank (or Mal'tsev-Prüfer rank).
- Type
- Chapter
- Information
- Groups St Andrews 2001 in Oxford , pp. 366 - 376Publisher: Cambridge University PressPrint publication year: 2003
- 1
- Cited by