The structure of Schur rings over cyclic groups

A graph $\Gamma$ is said to be unstable if for the direct product $\Gamma \times K_2$, $\mathrm{Aut}(\Gamma \times K_2)$ is not isomorphic to $\mathrm{Aut}(\Gamma) \times \mathbb{Z}_2$. We show that a connected and non-bipartite Cayley graph $\mathrm{Cay}(H,S)$ is unstable if and only if the set $S \times \{1\}$ belongs to a Schur ring over the group $H \times \mathbb{Z}_2$ satisfying certain properties. The S-rings with these properties are characterized if $H$ is a cyclic group of twice odd order. As an application, a necessary and sufficient condition is given for a connected and non-bipartite circulant graph of order $2p^e$ to be unstable, where $p$ is an odd prime and $e \ge 1$.

On the computational complexity of the theory of Abelian groups

In his famous monograph on permutation groups, H. Wielandt gives an example of a Schur ring over an elementary abelian group of order $p^2$ ($p>3$ is a prime), which is non-schurian, that is, it is the transitivity module of no permutation group. Generalizing this example, we construct a huge family of non-schurian Schur rings over elementary abelian groups of even rank.

Let G denote the finite cyclic group of order n. The problem of determining necessary and sufficient conditions for F to be the fixed point set of a smooth cyclic group action on some sphere has been solved when n is a prime power. P. A. Smith proved that F must be a Zn-homology sphere. If n is odd it is also known that F is unitary. L. Jones has shown that these conditions are also sufficient to realize F as the fixed point set of smooth cyclic group action on some sphere when n is a prime power Pl. In the general case it is known that F is a union of smooth manifolds, unitary if n is odd. And if Zn acts on some even dimensional sphere, then the Euler characteristic number of F is 2 if action is orientation-preserving and 0 if action is orientation-reversing. We may ask about possible restrictions on Pontryagin numbers of components of the fixed set F. If n is a prime power, the Pontryagin numbers of the fixed point set all vanish by the P. A. Smith theorem and the Hirzebruch signature theorem. It is natural to ask whether such restrictions hold for other types of smooth cyclic groups acting on spheres. In case where n is not a prime power, by work of R. Oliver [01] and its extention of A. Assadi [Asd] and K. Pawalowski [Pa 1-4], we can construct exotic actions on spheres such that Pontryagin classes of fixed point sets do not vanish. But in these examples the Pontryagin numbers all vanish because these actions bound group actions on disks. R. Schultz has shown in [S2] that if G is a cyclic group whose order is not a prime power, then there are smooth actions of G on spheres such that the fixed point sets have nonzero Pontryagin numbers provided the dimension of the fixed point set is greater than 16. Testing to see if the lower bound in dimensions is necessary. There are two possibilities. First, more sophisticated computations might make it possible to remove the restriction on dimensions. Second, there might be some unusual things happening in low dimensions (compare Ewing's result for Zp actions on spheres). Main Theorem is evidence for the first one.

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

Weights for symmetric and general linear groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

<p>As a powerful technique for holding relations in things, combinatorics has experienced rapidly development in the past century, particularly,enumeration of configurations, combinatorial design and graph theory. How-<br /> ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly,<br /> CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology,Euclidean geometry and differential geometry, non-solvable differential equations or classical mathematical systems with contradictions to mathematics,quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.</p>

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

Group theory is main topic modern algebra and group is also useful in many other fields. In this paper we show a special type relationship between group theory and number theory. In this paper will give type group of rational functions with coefficients as Fibonacci number with respect composition of mapping operation. We also show that is not only a group but also a cyclic group. In this represent a special relation between properties of group and rational functions with Fibonacci numbers coefficients. In this paper we gave a special type of recurrence relation sequence of rational functions with coefficients as Fibonacci numbers and also we proved the collection of all such rational function form a cyclic group with respect to the composition of function operation. In this represent a special relation between properties of group and rational functions with Fibonacci numbers coefficients. In this paper we gave a special type of recurrence relation sequence of rational functions with coefficients as Fibonacci numbers and also we proved the collection of all such rational function form a cyclic group with respect to the composition of function operation.

Let G be a group and let x, y ∈ G. If every element of G can be written as a word $${x^{{\alpha _1}}}{y^{{\alpha _2}}}{x^{{\alpha _3}}} \ldots {x^{{\alpha _{n - 1}}}}{y^{{\alpha _n}}}$$ (1) where αi ∈ ℤ, 1 ≤ i ≤ n, then we say that x and y generate G and that G is a 2-generator group. Although cyclic groups are 2-generator groups according to this definition we are only interested here in 2-generator groups which cannot be generated by a single element. Even among finite groups G many are not 2-generator groups; for example the abelian group of order 8 in which every element has order 2 cannot be 2-generated since given any pair of distinct non-trivial elements x, y there are only 4 words given by expressions of the form (1). However, many groups are 2-generated, in particular finite simple groups.

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.