scholarly journals On Automorphism Groups of Some Particular Groups

2016 ◽  
Vol 40 (2) ◽  
pp. 109-115
Author(s):  
Mohd Altab Hossain ◽  
Subrata Majumdar
Keyword(s):  
Automorphism Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Real Numbers ◽  
Cyclic Groups ◽  
Academy Of Sciences

The cyclic groups, the additive group Q of rational numbers and the additive group R of real numbers are sometimes very useful elements in many studies. In this paper, the authors concentrated their intuition in determining the structures of the automorphism groups of these useful groups in the light of previous works.Journal of Bangladesh Academy of Sciences, Vol. 40, No. 2, 109-115, 2016

scholarly journals Ext(Q,Z) is the additive group of real numbers

1969 ◽  
Vol 1 (3) ◽  
pp. 341-343 ◽  
Author(s):  
James Wiegold
Keyword(s):  
Cyclic Group ◽  
Additive Group ◽  
Rational Numbers ◽  
Infinite Subset ◽  
Real Numbers ◽  
Infinite Cyclic Group ◽  
Extra Effort ◽  
Uncountable Group ◽  
Countably Infinite

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

Right-Ordered Polycyclic Groups

1974 ◽  
Vol 17 (2) ◽  
pp. 175-178 ◽  
Author(s):  
Roberta Botto Mura
Keyword(s):  
Additive Group ◽  
Real Numbers ◽  
Ordered Groups ◽  
Polycyclic Groups

One of the features that make right-ordered groups harder to investigate than ordered groups is that their system of convex subgroups may fail to have the following property:(*) if C and C’ are convex subgroups of G and C’ covers C, then C is normal in C’ and C’/C is order-isomorphic to a subgroup of the naturally ordered additive group of real numbers.

scholarly journals Real order-automorphism groups

1981 ◽  
Vol 31 (3) ◽  
pp. 257-261 ◽  
Author(s):  
E. C. Weinberg
Keyword(s):  
Automorphism Groups ◽  
Real Numbers ◽  
Order Automorphism ◽  
Ordered Groups ◽  
Lattice Ordered Groups

AbstractBy using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.

An Editor Recalls Some Hopeless Papers

1998 ◽  
Vol 4 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Natural Numbers ◽  
Real Numbers ◽  
Diagonal Argument ◽  
The One ◽  
Academy Of Sciences ◽  
Main Message

§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank ([9] pp. 44f, 354).

Have You Read …?

1969 ◽  
Vol 62 (3) ◽  
pp. 220-221
Author(s):  
Philip Peak
Keyword(s):  
Real Number ◽  
Rational Number ◽  
Geometric Approach ◽  
Rational Numbers ◽  
Polynomial Equations ◽  
Real Numbers ◽  
Basic Principles ◽  
New Ideas

One of the basic principles we follow in our teaching is to relate new ideas with old ideas. Dr. Forbes has done just this in his article about extending the concept of rational numbers to real numbers. He points out how this extension cannot follow the same pattern as that of extensions positive to negative integers or from integers to rationals. If we look to a definition for motivating the extension we at best can only say, “Some polynomial equations have no rational number solutions and do have some real number solutions.” We might use least-upperbound idea, or we might try motivating through nonperiodic infinite decimals. However, Dr. Forbes rejects all of these and makes the tie-in through a geometric approach.

Automorphism groups of in nite meta-(locally cyclic) groups

2011 ◽  
Vol 41 (7) ◽  
pp. 613-628 ◽  
Author(s):  
HeGuo LIU ◽  
Jun LIAO
Keyword(s):  
Automorphism Groups ◽  
Cyclic Groups
Sign in / Sign up

Export Citation Format

Share Document