scholarly journals Simple graphs containing induced subgraphs whose automorphism groups are isomorphic to subgroups of a given finite group

1999 ◽  
Vol 206 (1-3) ◽  
pp. 205-212
Author(s):  
N. Sridharan ◽  
G. Ramachandran
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Induced Subgraphs ◽  
Simple Graphs

Nilpotent Partition-Inducing Automorphism Groups

1981 ◽  
Vol 33 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Identity Elements

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

scholarly journals Finite automorphism groups of laminated near-rings

1983 ◽  
Vol 26 (3) ◽  
pp. 297-306 ◽  
Author(s):  
K. D. Magill ◽  
P. R. Misra ◽  
U. B. Tewari
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Complex Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
The Other ◽  
Complex Polynomial ◽  
Infinite Groups ◽  
Nonsingular Matrices ◽  
A Finite Group

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

scholarly journals Finite groups as automorphism groups of orthocomplemented projective planes

1978 ◽  
Vol 25 (1) ◽  
pp. 19-24 ◽  
Author(s):  
Richard J. Greechie
Keyword(s):  
Finite Group ◽  
Projective Plane ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Projective Planes ◽  
Desarguesian Projective Plane ◽  
Absolute Point

AbstractA construction is given for a non-desarguesian projective plane P and an absolute-point free polarity on P such that the group of collineations of P which commute with the polarity is isomorphic to an arbitrary preassigned finite group.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

scholarly journals A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

2020 ◽  
Vol 22 (1) ◽  
pp. 38-47
Author(s):  
Dmitry S. Malyshev
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complete Classification ◽  
Induced Subgraphs ◽  
Simple Graphs ◽  
Forbidden Induced Subgraphs ◽  
The Third ◽  
Hereditary Class ◽  
Complexity Status

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

Diameters and automorphism groups of inclusion graphs over nilpotent groups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050097
Author(s):  
Shikun Ou ◽  
Dein Wong ◽  
Zhijun Wang
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Automorphism Groups ◽  
Undirected Graph ◽  
Nilpotent Groups ◽  
Dominating Sets ◽  
A Finite Group

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph that its vertices are all nontrivial subgroups of [Formula: see text], and in which two distinct subgroups [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. In this paper, we determine the diameter of [Formula: see text] when [Formula: see text] is nilpotent, and characterize the independent dominating sets as well as the automorphism group of [Formula: see text].

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

scholarly journals Automorphism groups of Beauville surfaces

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Gareth A. Jones
Keyword(s):  
Finite Group ◽  
Algebraic Surface ◽  
Automorphism Groups ◽  
Beauville Surfaces ◽  
A Finite Group ◽  
Beauville Surface

Abstract.A Beauville surface of unmixed type is a complex algebraic surface which is the quotient of the product of two curves of genus at least 2 by a finite group

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