Book contents
- Frontmatter
- Contents
- Contents of Volume II
- Introduction
- Radical rings and products of groups
- Homogeneous integral table algebras of degrees two, three and four with a faithful element
- A polynomial-time theory of black box groups I
- Totally and mutually permutable products of finite groups
- Ends and algebraic directions of pseudogroups
- On locally nilpotent groups with the minimal condition on centralizers
- Infinite groups in projective and symplectic geometry
- Non-positive curvature in group theory
- Group-theoretic applications of non-commutative toric geometry
- Theorems of Kegel-Wielandt type
- Singly generated radicals associated with varieties of groups
- The word problem in groups of cohomological dimension
- Polycyclic-by-finite groups: from affine to polynomial structures
- On groups with rank restrictions on subgroups
- On distances of multiplication tables of groups
- The Dade conjecture for the McLaughlin group
- Automorphism groups of certain non-quasiprimitive almost simple graphs
- Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators
- On p-pronormal subgroups of finite p-soluble groups
- On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata
- On the dimension of groups acting on buildings
- Dade's conjecture for the simple Higman-Sims group
- On the F*-theorem
- Covering numbers for groups
- Characterizing subnormally closed formations
- Symmetric words in a free nilpotent group of class 5
- A non-residually finite square of finite groups
On groups with rank restrictions on subgroups
Published online by Cambridge University Press: 05 August 2013
- Frontmatter
- Contents
- Contents of Volume II
- Introduction
- Radical rings and products of groups
- Homogeneous integral table algebras of degrees two, three and four with a faithful element
- A polynomial-time theory of black box groups I
- Totally and mutually permutable products of finite groups
- Ends and algebraic directions of pseudogroups
- On locally nilpotent groups with the minimal condition on centralizers
- Infinite groups in projective and symplectic geometry
- Non-positive curvature in group theory
- Group-theoretic applications of non-commutative toric geometry
- Theorems of Kegel-Wielandt type
- Singly generated radicals associated with varieties of groups
- The word problem in groups of cohomological dimension
- Polycyclic-by-finite groups: from affine to polynomial structures
- On groups with rank restrictions on subgroups
- On distances of multiplication tables of groups
- The Dade conjecture for the McLaughlin group
- Automorphism groups of certain non-quasiprimitive almost simple graphs
- Subgroups of the upper-triangular matrix group with maximal derived length and a minimal number of generators
- On p-pronormal subgroups of finite p-soluble groups
- On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata
- On the dimension of groups acting on buildings
- Dade's conjecture for the simple Higman-Sims group
- On the F*-theorem
- Covering numbers for groups
- Characterizing subnormally closed formations
- Symmetric words in a free nilpotent group of class 5
- A non-residually finite square of finite groups
Summary
Introduction
In this article we give a brief survey of recent work of the authors and others. We recall that if r is a fixed positive integer then a group G has (Prüfer) rank r if every finitely generated subgroup of G can be generated by r elements and r is the least such integer. Throughout this paper we shall say that a group G has rank r if, in the above sense, it has rank at most r. We shall be primarily concerned with those groups which have the property that certain of their subgroups have finite rank, although in the last section we discuss what might be thought of as the dual problem.
The types of question we have in mind are these: suppose that X is some class of groups and let G be a group in which every proper subgroup belongs to X. What can be said about G? Is G necessarily in X? Can we classify those groups G that are not in X? What if we relax the condition on G and only suppose that certain distinguished subgroups of G belong to X?
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- Groups St Andrews 1997 in Bath , pp. 237 - 247Publisher: Cambridge University PressPrint publication year: 1999
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