Exceptional group ring automorphisms for some metabelian groups

It is shown that exceptional automorphisms exist for semi-local group rings of groups involving extensions of certain generalized dihedral, semi-dihedral, or quaternion groups

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)∖{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.

A ring R is said to be a Baer (respectively, quasi-Baer) ring if the left annihilator of any nonempty subset (respectively, any ideal) of R is generated by an idempotent. It is first proved that for a ring R and a group G, if a group ring RG is (quasi-) Baer then so is R; if in addition G is finite then |G|–1 € R. Counter examples are then given to answer Hirano's question which asks whether the group ring RG is (quasi-) Baer if R is (quasi-) Baer and G is a finite group with |G|–1 € R. Further, efforts have been made towards answering the question of when the group ring RG of a finite group G is (quasi-) Baer, and various (quasi-) Baer group rings are identified. For the case where G is a group acting on R as automorphisms, some sufficient conditions are given for the fixed ring RG to be Baer.

Let M be a right module over a ring R and let G be a group. The set MG of all formal finite sums of the form ∑ g ∈ Gmgg where mg ∈ M becomes a right module over the group ring RG under addition and scalar multiplication similar to the addition and multiplication of a group ring. In this paper, we study basic properties of the RG-module MG, and characterize module properties of (MG)RG in terms of properties of MR and G. Particularly, we prove the module-theoretic versions of several well-known results on group rings, including Maschke’s Theorem and the classical characterizations of right self-injective group rings and of von Neumann regular group rings.

An exceptional automorphism is described for a semi-local group ring of a group of order 96. It is argued that there no smaller groups for which exceptional automorphisms exist.

In this paper, we study the structure of some Noetherian modules over group rings and deduce some statements regarding the structure of the groups involved. More precisely, we consider a module A over a group ring RG with the following property: A is a Noetherian RH-module for every subgroup H, which is not contained in the centralizer CG(A). If G is some generalized soluble group and R is a locally finite field or some Dedekind domain, we describe the structure of G/CG(A).

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

Let D = FlxF2x- xF" be a direct product of n free groups Ft, F2, , Fn, a an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed and T an infinite cyclic group.Let D x T be the semidirect product of D and T with respect to a.We prove that the Whitehead group of D x T and the projective class group of the integral group ring Z(D x T) are trivial.The second result implies that the projective class group of the integral group ring over the fundamental group of a surface is trivial.Let G be a group.We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by K0Z(G).Let a be an automorphism of G and let F be an infinite cyclic group.Then we denote by G x x F the semidirect product of G and F with respect to a.Let M be a connected 2-dimensional manifold and trx(M) the fundamental group of M. If M is open, then ttx(M) is a free group so that K0Z(ttx(M)) is trivial by a theorem of Bass (cf.[I]).Next, if Misasphereor a projective plane, then trx(M)=0 or T2 (cyclic group of order 2) and so K0Z(ttx(Mj) =0 (cf.[7, p. 419]).Now, if Mis closed and is not a sphere or projective plane, then Farrell-Hsiang[4] have shown that trx(M) is just the semidirect product F xxT, where F is a free group.The purpose of this paper is to show that K0Z(F xx T)=0 and so the projective class group of the integral group ring over the fundamental group of a surface is always trivial.In fact, we prove:

A ring R is called left p.q.-Baer if the left annihilator of a principal left ideal is generated, as a left ideal, by an idempotent. It is first proved that for a ring R and a group G, if the group ring RG is left p.q.-Baer then so is R; if in condition G is finite then |G|−1 ∈ R. Counterexamples are given to answer the question whether the group ring RG is left p.q.-Baer if R is left p.q.-Baer and G is a finite group with |G|−1 ∈ R. Further, RD∞ is left p.q.-Baer if and only if R is left p.q.-Baer.

A ring is right finite dimensional if it contains no infinite direct sum of right ideals.We prove that if a group G is finite, free abelian, or finitely generated abelian, then a ring R is right finite dimensional if and only if the group ring RG is right finite dimensional.A ring R is a self-injective cogenerator ring if Rn is injective and RR is a cogenerator in the category of unital right /{-modules; this means that each right unital -module can be embedded in a direct product of copies of R. Let G be a finite group where the order of G is a unit in R. Then the group ring RG is a selfinjective cogenerator ring if and only if R is a self-injective cogenerator ring.Additional applications are given.

We study configurations of 2-planes in that are combinatorially described by the Petersen graph. We discuss conditions for configurations to be locally Cohen–Macaulay and describe the Hilbert scheme of such arrangements. An analysis of the homogeneous ideals of these configurations leads, via linkage, to a class of smooth, general type surfaces in . We compute their numerical invariants and show that they have the unusual property that they admit (multiple) 7-secants. Finally, we demonstrate that the construction applied to Petersen arrangements with additional symmetry leads to surfaces with exceptional automorphism groups.

Preface Representations, Functors and Cohomology Cohomology and Representation Theory Jon F. Carlson 1. Introduction 2. Modules over p-groups 3. Group cohomology 4. Support varieties 5. The cohomology ring of a dihedral group 6. Elementary abelian subgroups in cohomology and representations 7. Quillen's dimension theorem 8. Properties of support varieties 9. The rank of the group of endotrivial modules Introduction to Block Theory Radha Kessar 1. Introduction 2. Brauer pairs 3. b-Brauer pairs 4. Some structure theory 5. Alperin's weight conjecture 6. Blocks in characteristic 7. Examples of fusion systems Introduction to Fusion Systems Markus Linckelmann 1. Local structure of finite groups 2. Fusion systems 3. Normalisers and centralisers 4. Centric subgroups 5. Alperin's fusion theorem 6. Quotients of fusion systems 7. Normal fusion systems 8. Simple fusion systems 9. Normal subsystems and control of fusion Endo-permutation Modules, a Guided Tour Jacques Th'evenaz 1. Introduction 2. Endo-permutation modules 3. The Dade group 4. Examples 5. The abelian case 6. Some small groups 7. Detection of endo-trivial modules 8. Classification of endo-trivial modules 9. Detection of endo-permutation modules 10. Functorial approach 11. The dual Burnside ring 12. Rational representations and an induction theorem 13. Classification of endo-permutation modules 14. Consequences of the classification An Introduction to the Representations and Cohomology of Categories Peter Webb 1. Introduction 2. The category algebra and some preliminaries 3. Restriction and induction of representations 4. Parametrization of simple and projective representations 5. The constant functor and limits 6. Augmentation ideals, derivations and H1 7. Extensions of categories and H2 Algebraic Groups and Finite Reductive Groups An Algebraic Introduction to Complex Reflection Groups Michel Brou'e Part I. Commutative Algebra: a Crash Course 1. Notations, conventions, and prerequisites 2. Graded algebras and modules 3. Filtrations: associated graded algebras, completion 4. Finite ring extensions 5. Local or graded k-rings 6. Free resolutions and homological dimension 7. Regular sequences, Koszul complex, depth Part II. Reflection Groups 8. Reflections and roots 9. Finite group actions on regular rings 10. Ramification and reflecting pairs 11. Characterization of reflection groups 12. Generalized characteristic degrees and Steinberg theorem 13. On the co-invariant algebra 14. Isotypic components of the symmetric algebra 15. Differential operators, harmonic polynomials 16. Orlik-Solomon theorem and first applications 17. Eigenspaces Representations of Algebraic Groups Stephen Donkin 1. Algebraic groups and representations 2. Representations of semisimple groups 3. Truncation to a Levi subgroup Modular Representations of Hecke Algebras Meinolf Geck 1. Introduction 2. Harish-Chandra series and Hecke algebras 3. Unipotent blocks 4. Generic Iwahori-Hecke algebras and specializations 5. The Kazhdan-Lusztig basis and the a-function 6. Canonical basic sets and Lusztig's ring J 7. The Fock space and canonical bases 8. The theorems of Ariki and Jacon Topics in the Theory of Algebraic Groups Gary M. Seitz 1. Introduction 2. Algebraic groups: introduction 3. Morphisms of algebraic groups 4. Maximal subgroups of classical algebraic groups 5. Maximal subgroups of exceptional algebraic groups 6. On the finiteness of double coset spaces 7. Unipotent elements in classical groups 8. Unipotent classes in exceptional groups Bounds for the Orders of the Finite Subgroups of G(k) Jean-Pierre Serre Lecture I. History: Minkowski, Schur 1. Minkowski 2. Schur 3. Blichfeldt and others Lecture II. Upper Bounds 4. The invariants t and m 5. The S-bound 6. The M-bound Lecture III. Construction of large subgroups 7. Statements 8. Arithmetic methods (k = Q) 9. Proof of theorem 9 for classical groups 10. Galois twists 11. A general construction 12. Proof of theorem 9 for exceptional groups 13. Proof of theorems 10 and 11 14. The case m = 1 Index

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.

A ring R is called right P-injective if every homomorphism from a principal right ideal of R to RR can be extended to a homomorphism from RR to RR. Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH, if f ∈ HomR(IR,RR), then there exists g ∈ HomR(RHR,RR) such that gI = f. Similarly, we also obtain equivalent characterizations of n-injective group rings and F-injective group rings.