Extra-Special Groups of Order 32 as Galois Groups

In this paper we give an explicit method for constructing every rational elliptic surface that has a four-torsion section. We use this construction to study the moduli space of birational equivalence classes of Galois covers that arise from rational elliptic surfaces with a relative dihedral group action. As a conclusion we find that the moduli space for such Galois covers whose Galois groups are the dihedral group of order eight is a nodal rational curve.

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.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is\nthe first step of Voevodsky's proof of this conjecture for arbitrary prime l)\nin a rather clear and elementary way. Assuming this conjecture, we construct a\n6-term exact sequence of Galois cohomology with cyclotomic coefficients for any\nfinite extension of fields whose Galois group has an exact quadruple of\npermutational representations over it. Examples include cyclic groups, dihedral\ngroups, the biquadratic group Z/2\\times Z/2, and the symmetric group S_4.\nSeveral exact sequences conjectured by Bloch-Kato, Merkurjev-Tignol, and Kahn\nare proven in this way. In addition, we introduce a more sophisticated version\nof the classical argument known as "Bass-Tate lemma". Some results about\nannihilator ideals in Milnor rings are deduced as corollaries.\n

Nonabelian 2-groups H containing a cyclic subgroup of index 2 are dihedral groups, generalized quaternion groups, quasidihedral groups and modular maximal-cyclic groups. Earlier the authors introduced the classes of piecewise quasiaffine transformations on an arbitrary nonabelian 2-group H with a cyclic subgroup of index 2. For the generalized group of quaternions of order 2 m we have obtained a complete classification of orthomorphisms, complete transformations and their left analogues in the class of piecewise quasiaffine transformations under consideration. This paper presents a similar classification for the remaining three groups (the dihedral group, the quasidihedral group and the modular maximal-cyclic group).

The author proves that every Abelian subgroup of rank in the Galois group of the algebraic closure of a rational function field is contained in a ramification subgroup, and also that the unramified Brauer group equals the unramified Brauer group defined in [2], §3, where is the quotient group .

Let [Formula: see text] be an elliptic curve. Let [Formula: see text] denote the compositum of all fields with Galois group [Formula: see text] over [Formula: see text]. We classify the possible torsion subgroups of [Formula: see text] over [Formula: see text] for [Formula: see text] not divisible by 3 or 4, where [Formula: see text] is the dihedral group on a regular [Formula: see text]-gon. We also obtain upper bounds for the torsion of [Formula: see text] over [Formula: see text] for various semi-direct products [Formula: see text] by way of classifications of torsion over [Formula: see text].

Let g be a univariate separable polynomial of degree n with coefficients in a computable field K and let (α1, . . . , αn) be an n-tuple of its roots in an algebraic closure K of K. Obtaining an algebraic representation of the splitting field K(α1, . . . , αn) of g is a question of first importance in effective Galois theory. For instance, it allows us to manipulate symbolically the roots of g. In this paper, we focus on the computation of the splitting field of g when its Galois group is a dihedral group. We provide an algorithm for this task which returns a triangular set encoding the relations ideal of g which has degree 2n since the Galois group of g is dihedral. Our algorithm starts from a factorization of g in K[X]/ and constructs the searched triangular set by performing n2 computations of normal forms modulo an ideal of degree 2n.

Let X be a subgroup of a group Y. The \\emphinterval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (N K/k K*, k*) contains a finite number of norm groups for any finite extension K of a 𝔭-adic number field k. In the present work we investigate the number of norm groups in the interval (N K/k K*, k*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A n or S n , then the interval (N K/k K*, k*) contains only the obvious two norm groups. Also, the interval (N K/k K*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2𝔫-\\emphextensions. In the present work we generalize the main theorem to non-Galois 2𝔫-extensions K/k, and determine some subintervals of (N K/k K*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois 2-extension K/k with the Galois group that either contains an element of order 8 or contains the quaternion group Q8 of order 8, or Q8 is a homomorphic image of the Galois group.

<abstract> Let $ G $ be a finite group. The reduced power graph of $ G $ is the undirected graph whose vertex set consists of all elements of $ G $, and two distinct vertices $ x $ and $ y $ are adjacent if either $ \langle x\rangle \subset \langle y\rangle $ or $ \langle y\rangle \subset \langle x\rangle $. In this paper, we show that the reduced power graph of $ G $ is perfect and characterize all finite groups whose reduced power graphs are split graphs, cographs, chordal graphs, and threshold graphs. We also give complete classifications in the case of abelian groups, dihedral groups, and generalized quaternion groups. </abstract>

Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $\langle x \rangle \subset \langle y \rangle$ or $\langle y \rangle \subset \langle x \rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $\mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

The strong metric dimension of the power graph of a finite group

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.

Perfect Isometries for Principal Blocks with Abelian Defect Groups and Elementary Abelian 2-Inertial Quotients

Blocked triangular patterns and their symmetry groups