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 the F*-theorem
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
With respect to the structure of torsion subgroups of integral group rings the following conjecture due to Zassenhaus has been over the last twenty years in the middle of the research. The conjecture may be stated as follows.
(ZC) Let ℤG be the integral group ring of the finite group G. Denote the units of augmentation 1 by V(ℤG) and let H be a subgroup of V(ℤG) of the same order as G. Then there exists a central automorphism σ of ℤG with σ(G) = H.
The conjecture is also of interest for more general coefficient rings than ℤ. We say that (ZC) holds for a group ring RG, if the content of the conjecture holds in RG. It has been shown by Roggenkamp and Scott that (ZC) does not hold for any finite group [9], [13]. But, if (ZC) is true, it gives a strong answer to the isomorphism problem of integral group rings. For a recent survey about the Zassenhaus conjecture and related questions we refer to [6]. For the newest developments with respect to the isomorphism problem see [2], [3] rsp.
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- Groups St Andrews 1997 in Bath , pp. 346 - 352Publisher: Cambridge University PressPrint publication year: 1999
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