On certain groups with Chernikov group of automorphisms

2014 ◽  
Vol 22 ◽  
pp. 72
Author(s):  
O.O. Pypka
Keyword(s):  
Automorphism Group ◽  
Factor Group ◽  
Chernikov Group ◽  
Group Of Automorphisms ◽  
Schur’S Theorem

We obtained automorphic analogue of Schur’s theorem for the case when an arbitrary subgroup A of automorphism group Aut(G) of a group G and the factor-group of a group G modulo A-center are Chernikov groups.

Complete set of Genera of Compact Riemann Surfaces with reference to the point Group of Sulphur (S8) Molecule

2020 ◽  
Vol 32 (7) ◽  
pp. 88-92
Author(s):  
RAFIQUL ISLAM ◽  
◽  
CHANDRA CHUTIA ◽  
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Point Group ◽  
Finite Point ◽  
Group Of Automorphisms ◽  
Complete Set ◽  
Compact Riemann Surfaces ◽  
Group Of Symmetries

In this paper we consider the group of symmetries of the Sulphur molecule (S8 ) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u\upsilon/u2= \upsilon8 = (u\upsilon)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which D16 acts as a group of automorphisms as follows: D16 the group of symmetries of the sulphur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers \lambda and \mu such that \lambda \leq 1 and \mu \geq 1 and g=\lambda +8\mu (\geq2) , \mu\geq |\lambda|

scholarly journals Automorphisms of compact non-orientable Riemann surfaces

1971 ◽  
Vol 12 (1) ◽  
pp. 50-59 ◽  
Author(s):  
D. Singerman
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Riemann Surfaces ◽  
Group Of Automorphisms ◽  
Groups Of Automorphisms ◽  
Definition Of ◽  
Orientable Surfaces

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

scholarly journals THE GROUP OF AUTOMORPHISMS OF A 3-GENERATED 2-GROUP OF INTERMEDIATE GROWTH

2004 ◽  
Vol 14 (05n06) ◽  
pp. 667-676 ◽  
Author(s):  
R. I. GRIGORCHUK ◽  
S. N. SIDKI
Keyword(s):  
Automorphism Group ◽  
Group Of Automorphisms ◽  
Intermediate Growth ◽  
Infinite Rank

The automorphism group of a 3-generated 2-group G of intermediate growth is determined and it is shown that the outer group of automorphisms of G is an elementary abelian 2-group of infinite rank.

Automorphism Groups of the Imprimitive Complex Reflection Groups II

2011 ◽  
Vol 18 (02) ◽  
pp. 315-326
Author(s):  
Li Wang
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Automorphism Groups ◽  
Reflection Group ◽  
Reflection Groups ◽  
Group Of Automorphisms ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Scalar Multiple

We prove that the automorphism group Aut (m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself.

A faithful polynomial representation of Out F3

1989 ◽  
Vol 106 (2) ◽  
pp. 207-213 ◽  
Author(s):  
James McCool
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Polynomial Ring ◽  
Outer Automorphism ◽  
Faithful Representation ◽  
Partial Answer ◽  
Polynomial Representation ◽  
Group Of Automorphisms ◽  
Outer Automorphism Group ◽  
Torsion Free

Let Fn be a free group of rank n and let Out Fn be its outer automorphism group. The main result of this paper is that Out F3 has a faithful representation as a group of automorphisms of the polynomial ring in seven variables over the integers. This extends a similar result for n = 2 (see Helling [3], Horowitz [5] and Rosenberger [12]), and provides a partial answer to a conjecture attributed in [5] to W. Magnus. As an application of the special nature of the representing polynomials, we obtain our second result, that Out F3 is virtually residually torsion-free nilpotent.

On the structure and some numerical properties of subgroups and factor-groups defined by automorphism groups

2015 ◽  
Vol 14 (05) ◽  
pp. 1550075
Author(s):  
Leonid A. Kurdachenko ◽  
Javier Otal ◽  
Alexander A. Pypka
Keyword(s):  
Commutator Subgroup ◽  
Automorphism Groups ◽  
Factor Group ◽  
Group Of Automorphisms ◽  
Special Rank ◽  
The Relationship

A group of automorphisms A of a group G defines the A-center CG(A) of G and the A-commutator subgroup [G, A] of G that naturally extend the ordinary center and the commutator subgroup of G. In this paper we study the relationship between the factor group G/CG(A) and the subgroup [G, A] when A has finite special rank and Inn (G) ≤ A.

The upper central series of the maximal normal 𝑝-subgroup of a group of automorphisms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Alicia Aviño ◽  
Phill Schultz ◽  
Marcos Zyman
Keyword(s):  
Automorphism Group ◽  
Nilpotency Class ◽  
Central Series ◽  
Group Of Automorphisms

Abstract Let 𝐺 be a bounded abelian 𝑝-group, with automorphism group Aut ⁡ ( G ) \operatorname{Aut}(G) . Whenever 𝐺 satisfies certain conditions, we determine the upper central series and nilpotency class of the maximal normal 𝑝-subgroup of Aut ⁡ ( G ) \operatorname{Aut}(G) .

scholarly journals Cubic Edge-Transitive Bi-$p$-Metacirculants

10.37236/6417 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yan-Li Qin ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Group Of Automorphisms ◽  
Group A ◽  
Semisymmetric Graphs ◽  
Edge Transitive

A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.

scholarly journals On the virtual automorphism group of a minimal flow

2020 ◽  
pp. 1-15
Author(s):  
JOSEPH AUSLANDER ◽  
ELI GLASNER
Keyword(s):  
Automorphism Group ◽  
Minimal Flow ◽  
Group Of Automorphisms ◽  
The Relationship

We introduce the notions ‘virtual automorphism group’ of a minimal flow and ‘semiregular flow’ and investigate the relationship between the virtual and actual group of automorphisms.

Sign in / Sign up

Export Citation Format

Share Document