Exceptional group ring automorphisms for some metabelian groups, ii

Let H be a generalized dihedral, semi-dihedral, quaternion, or modular group, and let A = (u, v, w) be a product of three odd order cyclic groups, with (|v|,|w|) = 1. For R a semi-local Dedekind domain of characteristic 0 in which no prime divisor of |H|.|A| is invertible, we prove that there is a semi-direct product G = H × A such that the group ring RG has an exceptional automorphism, i.e. provides a counter-example to a well-known conjecture of Zassenhaus on automorphisms of 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.

Perfect state transfer on bi-Cayley graphs over abelian groups

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.

The Paul Erdős and Andras Gyarfas conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order \(p^3\).

Grothendieck groups of dihedral and quaternion group rings

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

The holomorph of a group G is Norm B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.

In chemistry, the molecular structure can be represented as a graph. Based on the information from the graph, its characterization can be determined by computing the topological index. Topological index is a numerical value that can be computed by using some algorithms and properties of the graph. Meanwhile, the non-commuting graph is a graph, in which two distinct vertices are adjacent if and only if they do not commute, where it is made up of the non-central elements in a group as a vertex set. In this paper, the Szeged index of the non-commuting graph of some finite groups are computed. This paper focuses on three finite groups which are the quasidihedral groups, the dihedral groups, and the generalized quaternion groups. The construction of the graph is done by using Maple software. In finding the Szeged index, some of the previous results and properties of the graph for the quasidihedral groups, the dihedral groups, and the generalized quaternion groups are used. The generalisation of the Szeged index of the non-commuting graph is then established for the aforementioned groups. The results are then applied to find the Szeged index of the non-commuting graph of ammonia molecule.

Let G be a finite group and let e be its identity element. The intersection power graph of G is the undirected graph with vertex set G, in which two distinct vertices x and y are adjacent if either 〈 x 〉 ∩ 〈 y 〉 ≠ { e } or one of x and y is e. In this paper, we first compute the strong metric dimension of the intersection power graph of a cyclic group, a dihedral group, and a generalized quaternion group. We also characterize all finite groups G whose intersection power graph has strong metric dimension | G | − 2 . Then, we give the sharp upper and lower bounds for the metric dimension of the intersection power graph of a finite group. As applications, we obtain the metric dimension of the intersection power graph of a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n -fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a ℤ 2 -extension of a cyclic group, including generalized quaternion and semi-dihedral groups.

We propose a dimension-reducing transformation on Group Ring Learning with Errors (GRLWE) samples. We exhibit an efficiently computable isomorphism which takes samples defined over the group rings used in the construction of GRLWE to twice as many samples defined over matrix rings, in half the dimension. This is done by composing two maps: the first map is a transformation showing that the group rings used are orders in central simple algebras, and the second map takes the obtained central simple algebra to a matrix ring. When combined with lattice reduction on the resulting matrix samples, this gives an attack on the GRLWE problem. We extend this attack to other groups proposed for cryptographic use by the creators of GRLWE, and display some numerical results quantifying the effects of the transformation, using the `Lattice Estimator'. We then give a family of groups from which GRLWE-style group rings can be constructed which are immune to our attack, namely the generalized quaternion groups. Finally, we discuss the merits and vulnerabilities of a number of different forms of structured LWE.

This is a continuation of our study [3] of a family of projective modules over Q4n , the generalized quaternion (binary dihedral) group of order 4n. Our approach is constructive. Whenever n 7 is odd, this work provides examples of stably free nonfree modules of rank 1, which are then used to construct exotic algebraic 2‐ complexes relevant to Wall’s D(2)‐problem. While there are examples of stably free nonfree modules for many infinite groups G , there are few actual examples for finite groups. This paper offers an infinite collection of finite groups with stably free nonfree modules P , given as ideals in the group ring. We present a method for constructing explicit stabilizing isomorphisms W ZG ZGaP ZG described by 2 2 matrices. This makes the subject accessible to both theoretical and computational investigations, in particular, of Wall’s D(2)‐problem. 16D40, 19A13, 57M20; 55P15

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups include the dihedral groups, the non-abelian groups of order pq, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.

Let K[G] denote the group ring of G over the field K. In this note we characterize those group rings in which all left ideals are right ideals. Let R be a ring. We say that R is l.i.r.i. if every left ideal is a right ideal. A ring is l.a.r.i. if every left annihilator is a right ideal. Our notation follows that of [2]. The main results are THEOREM I. Let K be a field and let G be a nonabelian periodic group. Then if K[G] is l.a.r.i. one of the following occurs (i) Char K = 0 and G is a Hamiltonian group such that for each odd exponent, n, of G the quaternion algebra over the field K(4), where 4 is a primitive nth root of unity, is a division ring. (ii) Char K = 2 and K does not contain any primitive cube root of unity. Moreover G = Q x A, where Q is the quaternion group of order 8 and A is abelian in which each element has odd order and if n is an exponent for A, the least integer m > 1 satisfying 2m -1 (mod n) is odd. Conversely if K[G] satisfies either (i) or (ii), then K[G] is l.i.r.i. and, in particular, it is l.a.r.i. Observe that if char K > 2 and G is periodic, then K[GI is l.a.r.i. if and only if G is abelian. THEOREM II. Let K[G] denote the group ring over a nonabelian group. Then the following are equivalent (i) K[G] is l.i.r.i. (ii) G is locally finite and if a,f8 E K[GI with a,8 = 0, then /3a = 0. (iii) G is locally finite and K[G] is l.a.r.i. If we combine the above theorems we get necessary and sufficient conditions for K[GI to be l.i.r.i. By using the antiautomorphism of K[G] given by ,xeG k.x H* IeG kxxwe see that K[G] is l.i.r.i. (l.a.r.i.) if and only if K[G] is r.i.l.i. (r.a.l.i.). Received by the editors February 2, 1978. AMS (MOS) subject classifications (1970). Primary 16A26.