On generic cyclic polynomials of odd prime degree

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. It follows from these constructions that there is a group-based complete Latin square of order $n$ if and only if $n \in \{ 1,2,4\}$ or there is a non-abelian group of order $n$.

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

We now deal with problems of p odd similar to those of Chapter V for p = 2. We lead off in section 43 with differentiable, orientation preserving actions of (Z p ) k , p an odd prime, on closed oriented manifolds V n . The primary aim is to give existence theorems for stationary points of such actions. Our interest in such problems has been aroused by the work of Borel [6, 9], although we attack the problem from a different point of view. An example of a corollary of our results is that if V n has one of its Pontryagin numbers not divisible by p then the action has a stationary point. We go on to note that if a toral group acts on V n without stationary points then [V n ] represents a torsion element of Ω n .

On the Birch-Tate conjecture for cyclic number fields

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending on who starts. These invariants could be infinite, thus we start by giving sufficient conditions to have infinite game distinguishing numbers. We also show that for graphs with cyclic automorphisms group of prime odd order, both game invariants are finite. After that, we define a class of graphs, the involutive graphs, for which the game distinguishing number can be quadratically bounded above by the classical distinguishing number. The definition of this class is closely related to imprimitive actions whose blocks have size $2$. Then, we apply results on involutive graphs to compute the exact value of these invariants for hypercubes and even cycles. Finally, we study odd cycles, for which we are able to compute the exact value when their order is not prime. In the prime order case, we give an upper bound of $3$.

The aim of this series of papers is to study cyclic extensions of the field of rational numbers of odd prime power degree. This paper, which is the third of this series, concentrates on the situation where some of the ramified primes are (partially) decomposed.

The paper investigates upper bounds on the period length of functional continued fractions for key elements of hyperelliptic fields over number fields. In the case when the hyperelliptic field is given by a polynomial of odd degree, the finite period length is trivially estimated from above twice the power of the fundamental 𝑆-unit. A more interesting and complicated case is when the hyperelliptic field is given by a polynomial of even degree. In 2019 V.P. Platonov andG.V. Fedorov for hyperelliptic fields ℒ = Q(𝑥)(√𝑓), deg 𝑓 = 2𝑔 +2, over the field Q of rational numbers the exact interval of values 𝑠 ∈ Z is found such that the continued fractions of elements of the form√𝑓/𝑥𝑠 ∈ ℒ∖Q(𝑥) are periodic. In this article, we find a generalization of this result for an arbitrary field as a field of constants. Based on this result, sharp upper estimates for the lengths of the periods are found functional continued fractions of elements of hyperelliptic fields over number fields 𝐾, depending only on the genus 𝑔 of the hyperelliptic field, the degree of extension 𝑘 = [𝐾 : Q] and order 𝑚 of the Jacobian torsion subgroup of the corresponding hyperelliptic curve.

Let $\{ {T_j ( X )} \}_{j = 0}^n $ be the Chebyshev polynomials of the first kind, where $T_j (x)$ is the polynomial of jth degree. It is shown that there exists a unique best approximation of $T_1 (x)$ with respect to the linear space spanned by polynomials of odd degree $\geqq 3$, which is also a best approximation of $T_1 (x)$ with respect to the linear space spanned by $\{ {T_j (x)} \}_{j = 0,j \ne 1}^n $. If $n = 4k$ or $4k - 1$, the extreme points of the deviation of $T_1 (x)$ from its best approximation are $2k$ in number, whereas if $n = 4k + 1$ or $4k + 2$, this number is $2k + 2$. The extreme point sets are explicitly obtained.

Data reduction using the least square method

We present a fully secure ciphertext-policy attribute-based encryption (CP-ABE) scheme in prime order group by using a tool which can convert composite order paring-based encryption scheme into prime order one. Although, a fully secure CP-ABE scheme in prime order group has been proposed recently, our scheme needs less master secret key size and can be seen as a different version of it. Besides, the sketch of our scheme's security proof is also given.

We present a new encoding framework for predicate encryption (PE) in prime order groups. Our framework captures a broader range of adaptively secure PE schemes by allowing PE schemes to have more flexible (i.e., non-linear) structures. The existing works dealing with adaptively secure PE schemes in prime order groups require strict structural restrictions on PE schemes. In particular, the exponents of public keys and master secret keys of the PE schemes, which are referred to as common variables, must be linear. In this paper, we introduce a modular approach which includes non-linear common variables in PE schemes. First, we formalize non-linear structures by improving Attrapadung’s pair encoding framework (Eurocrypt’14). Then, we provide a generic compiler that incorporates encodings under our framework to PE schemes in prime order groups. Notably, we prove the security of our compiler by introducing a new technique that decomposes common variables into two types and makes one of them shared between semi-functional and normal spaces on processes of the dual system encryption. As instances of our new framework, we introduce new attribute-based encryption schemes supporting non-monotone access structures, namely non-monotonic ABE. Our new schemes are adaptively secure in prime order groups and have either short ciphertexts (in the case of KP-ABE) or short keys (in the case of CP-ABE).

1 Introduction and Preliminaries All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Itô [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Itô's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is π-quasinormal in G if H permutes with every Sylow subgroup of G. More recently, Shirong [9] proved that if the finite group G possesses a normal subgroup N of odd order such that G/N is supersoluble, and if, for each Sylow subgroup P of N, every minimal subgroup of P is normal in NG (P), then G is supersoluble. In [10] and [11] Yokoyama extends the results of Itô and Buckley for soluble groups using formation theory to generalize the notion of centrality.

In Theorem B of [1], a very general Theorem of Hall-Higman type for groups of odd order was obtained. We mentioned there the possibility to drop the hypothesis (]A], r - 1) = 1 in the statement. The purpose of this note is to do it so. We give also an example showing that the assumption of r being odd is necessary. Our proof will consist into an extension of Theorem A of [1] to the case in which a group of odd order acts on an extraspecial r-group not necessarily centralizing the centre. By Theorem A of [1], we may suppose that r _-> 7. We use some auxiliary results of [l], too. We begin with two preliminary lemmas: Lemma 1. Let G be a group of odd order such that every Sylow subgroup of the Fitting subgroup F(G) of G is the central product of a cyclic group and an extraspecial group of prime exponent. Suppose that V is a KG-module, K a field, and [V,y] = V /f 1 ~ y ~ Z(F(G)). If x is a nontrivial element of G, then

In 1980, McCaughan and Stonehewer showed that a finite soluble group in which every subnormal subgroup has defect at most two has derived length at most nine and Fitting length at most five, and gave an example of derived length five and Fitting length four. In 1984 Casolo showed that derived length five and Fitting length four are best possible bounds.In this paper we show that for groups of odd order the bounds can be improved. A group of odd order with every subnormal subgroup of defect at most two has derived and Fitting length at most three, and these bounds are best possible.