Some New Groups which are not CI-groups with Respect to Graphs

In this paper, we prove that the Cayley digraph Γ = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(Γ), a graph constructed from Γ by using the method presented in the paper, is primitive on the vertices set V(A(Γ). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq2 are normal, where p and q are distinct odd primes.

In this article we examine conditions for the appearance or nonappearance of the twoextra-special 2-groups of order 32 as Galois groups over a fieldFof characteristic not 2. The groups in question are the central productsDDof two dihedral groups of order 8, andDQof a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups overFimplies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability ofDDandDQ. We conclude by providing an explicit construction of field extensions with Galois groupDD.

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.

An independent set of a graph is a set of pairwise non-adjacent vertices. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate. The noncommuting graph is defined as a graph whose vertex set is non-central elements in which two vertices are adjacent if and only if they do not commute. In this research, the independence polynomial of the conjugate graph and noncommuting graph are found for three nonabelian groups of order at most eight.

A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.

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 $n$.

Recently, it was proved that every p-Schur ring over an abelian group of order <TEX>$p^3$</TEX> is Schurian. In this paper, we prove that every commutative p-Schur ring over a non-abelian group of order <TEX>$p^3$</TEX> is Schurian.

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this thesis, we characterize D_4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D_4-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D_4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D_4 is a local Oort group for the prime 2. Furthermore, we characterize Q_8-extensions of fields of characteristic two, determine the ramification breaks of (suitable) Q_8-extensions of complete discrete valuation fields of characteristic two, and, by solving the local lifting problem in the negative for a family of Q_8-extensions of complete discrete valuation fields of characteristic two with algebraically closed residue field, show that neither Q_8 nor SL_2(Z/3Z) is an almost local Oort group for the prime 2.

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.

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

Blocked triangular patterns and their symmetry groups

On some 10-arcs for deriving the minimum order for complete arcs in small projective planes

: A proof is presented of the following theroem: Let X be a compact cohomology n-manifold over Z, the ring of integers, with H*(X;Z) equal to H*(S(n);Z) and let G equal SO(3), the rotational of the euclidean 3-space, act on X with B equal to (n-2), where B is the union of all singular orbits of dimension Z. Then D does not equal zero and one of the following occurs: (1) n equals one and G acts trivially on X; (2) n is greater than or equal to 4 and for every Y belonging to D,G(Y) is a dihedral group of order 4 (Y is an element of X); or (3) n is greater than or equal to 5, and for every Y belonging to U, G(Y) is the identity group.

An m-Cayley graph $$\Gamma $$ over a group G is defined as a graph which admits G as a semi-regular group of automorphisms with m orbits. This generalises the notions of a Cayley graph (where $$m = 1$$ ) and a bi-Cayley graph (where $$m = 2$$ ). The m-Cayley graph $$\Gamma $$ over G is said to be normal if G is normal in the automorphism group $$\mathrm{Aut}(\Gamma )$$ of $$\Gamma $$ , and core-free if the largest normal subgroup of $$\mathrm{Aut}(\Gamma )$$ contained in G is the identity subgroup. In this paper, we investigate properties of symmetric m-Cayley graphs in the special case of valency 3, and use these properties to develop a computational method for classifying connected cubic core-free symmetric m-Cayley graphs. We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31–41 2005). Using our classification method, we give a new proof of the fact that there are exactly 15 connected cubic core-free symmetric Cayley graphs, two of which are Cayley graphs over non-abelian simple groups. We also show that there are exactly 109 connected cubic core-free symmetric bi-Cayley graphs, 48 of which are bi-Cayley graphs over non-abelian simple groups, and that there are 1, 6, 81, 462 and 3267 connected cubic core-free 1-arc-regular 3-, 4-, 5-, 6- and 7-Cayley graphs, respectively.

Integral Cayley graphs over dicyclic group