scholarly journals GBRDs with Block Size Three over 2-Groups, Semi-Dihedral Groups and Nilpotent Groups

10.37236/519 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. Julian R. Abel ◽  
Diana Combe ◽  
Adrian M. Nelson ◽  
William D. Palmer
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Nilpotent Groups ◽  
Dihedral Groups ◽  
Even Order ◽  
Generalized Bhaskar Rao Design

There are well known necessary conditions for the existence of a generalized Bhaskar Rao design over a group $\mathbb{G}$, with block size $k=3$. We prove that they are sufficient for nilpotent groups $\mathbb{G}$ of even order, and in particular for $2$-groups. In addition, we prove that they are sufficient for semi-dihedral groups.

scholarly journals The Existence of FGDRP$(3,g^u)'$s

10.37236/123 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jie Yan ◽  
Chengmin Wang
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Main Diagonal ◽  
Parallel Class ◽  
Uniform Frame ◽  
Index 2

By an FGDRP$(3,g^u)$, we mean a uniform frame $(X,\cal G,\cal A)$ of block size 3, index 2 and type $g^u$, where the blocks of $\cal{A}$ can be arranged into a $gu/3\times gu$ array. This array has the properties: (1) the main diagonal consists of $u$ empty subarrays of sizes $g/3\times g$; (2) the blocks in each column form a partial parallel class partitioning $X \setminus G$ for some $G\in \cal G$, while the blocks in each row contain every element of $X \setminus G$ $3$ times and no element of $G$ for some $G\in \cal{G}$. The obvious necessary conditions for the existence of an FGDRP$(3,g^u)$ are $u\geq 5$ and $g\equiv 0$ (mod 3). In this paper, we show that these conditions are also sufficient with the possible exceptions of $(g,u)\in \{(6,15),(9,18),(9,28),(9,34),(30,15)\}$.

Affine Completeness of Generalised Dihedral Groups

2006 ◽  
Vol 49 (3) ◽  
pp. 347-357 ◽  
Author(s):  
Jürgen Ecker
Keyword(s):  
Direct Product ◽  
Nilpotent Groups ◽  
Necessary Condition ◽  
Dihedral Groups ◽  
Complete Group ◽  
Affine Completeness ◽  
Affine Complete

AbstractIn this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every k ∈ N the k-affine complete generalised dihedral groups. We find that the direct product of a 1-affine complete group with itself need not be 1-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for 2-affine completeness.

scholarly journals On extendability of Cayley graphs

Filomat ◽  
2009 ◽  
Vol 23 (3) ◽  
pp. 93-101 ◽  
Author(s):  
Stefko Miklavic ◽  
Primoz Sparl
Keyword(s):  
Abelian Group ◽  
Cayley Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Cayley Graphs ◽  
Dihedral Groups ◽  
Odd Order ◽  
Even Order

A connected graph ? of even order is n-extendable, if it contains a matching of size n and if every such matching is contained in a perfect matching of ?. Furthermore, a connected graph ? of odd order is n1/2-extendable, if for every vertex v of ? the graph ? - v is n-extendable. It is proved that every connected Cayley graph of an abelian group of odd order which is not a cycle is 1 1/2-extendable. This result is then used to classify 2-extendable connected Cayley graphs of generalized dihedral groups.

scholarly journals Some Notes on Relative Commutators

2020 ◽  
Vol 2 (2) ◽  
pp. 65-70
Author(s):  
Masoumeh Ganjali ◽  
Ahmad Erfanian
Keyword(s):  
Nilpotent Group ◽  
Cyclic Group ◽  
Commutator Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Perfect Group ◽  
Group A ◽  
Even Order ◽  
Definition Of

Let G be a group and α ϵ Aut(G).  An α-commutator of elements x, y ϵ G is defined as [x, y]α = x-1y-1xyα. In 2015, Barzegar et al. introduced an α-commutator of elements of G and defined a new generalization of nilpotent groups by using the definition of α-commutators which is called an α-nilpotent group. They also introduced an α-commutator subgroup of G, denoted by Dα(G) which is a subgroup generated by all α-commutators. In 2016, an α-perfect group, a group that is equal to its α-commutator subgroup, was introduced by authors of this paper and the properties of such group was investigated. They proved some results on α-perfect abelian groups and showed that a cyclic group G of even order is not α-perfect for any α ϵ Aut(G). In this paper, we may continue our investigation on α-perfect groups and in addition to studying the relative perfectness of some classes of finite p-groups, we provide an example of a non-abelian α-perfect 2-group.

scholarly journals Group Divisible Designs with Two Associate Classes and

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Nittiya Pabhapote ◽  
Narong Punnim
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Group Divisible Designs ◽  
Group Divisible

The original classiffcation of PBIBDs defined group divisible designs GDD() with . In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with .

Image Compression Using Vector Quantization with Variable Block Size Division

2010 ◽  
Vol 130 (8) ◽  
pp. 1431-1439 ◽  
Author(s):  
Hiroki Matsumoto ◽  
Fumito Kichikawa ◽  
Kazuya Sasazaki ◽  
Junji Maeda ◽  
Yukinori Suzuki
Keyword(s):  
Image Compression ◽  
Vector Quantization ◽  
Block Size ◽  
Variable Block

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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