scholarly journals Delta-invariants for Fano varieties with large automorphism groups

2020 ◽  
Vol 31 (10) ◽  
pp. 2050077
Author(s):  
Aleksei Golota
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Fano Variety ◽  
Fano Varieties ◽  
Connected Subgroup ◽  
Closed Connected Subgroup ◽  
A Finite Group ◽  
Large Groups

For a variety [Formula: see text], a big [Formula: see text]-divisor [Formula: see text] and a closed connected subgroup [Formula: see text] we define a [Formula: see text]-invariant version of the [Formula: see text]-threshold. We prove that for a Fano variety [Formula: see text] and a connected subgroup [Formula: see text] this invariant characterizes [Formula: see text]-equivariant uniform [Formula: see text]-stability. We also use this invariant to investigate [Formula: see text]-equivariant [Formula: see text]-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of [Formula: see text] being a finite group.

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.

Groups of real genus p + 1, where p is prime

2014 ◽  
Vol 14 (03) ◽  
pp. 1550040
Author(s):  
Coy L. May
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Klein Surface ◽  
The Real ◽  
A Finite Group ◽  
Large Groups

Let G be a finite group. The real genusρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We classify the large groups of real genus p + 1, that is, the groups such that |G| ≥ 3(g - 1), where the genus action of G is on a bordered surface of genus g = p + 1. The group G must belong to one of four infinite families. In addition, we determine the order of the largest automorphism group of a surface of genus g for all g such that g = p + 1, where p is a prime.

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$\mathbb{Z}$.

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].

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

More on the OD-Characterizability of a Finite Group

2011 ◽  
Vol 18 (04) ◽  
pp. 663-674 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Prime Graphs ◽  
A Finite Group ◽  
P 53 ◽  
Number Of Authors

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut (O7(3)) and Aut (S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

Burnside rings of fusion systems and their unit groups

2020 ◽  
Vol 23 (4) ◽  
pp. 709-729
Author(s):  
Jamison Barsotti ◽  
Rob Carman
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Unit Group ◽  
Fusion System ◽  
Burnside Ring ◽  
Fusion Systems ◽  
A Finite Group ◽  
Burnside Rings

AbstractFor a saturated fusion system {\mathcal{F}} on a p-group S, we study the Burnside ring of the fusion system {B(\mathcal{F})}, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring {B(S)}. We give criteria for an element of {B(S)} to be in {B(\mathcal{F})} determined by the {\mathcal{F}}-automorphism groups of essential subgroups of S. When {\mathcal{F}} is the fusion system induced by a finite group G with S as a Sylow p-group, we show that the restriction of {B(G)} to {B(S)} has image equal to {B(\mathcal{F})}. We also show that, for {p=2}, we can gain information about the fusion system by studying the unit group {B(\mathcal{F})^{\times}}. When S is abelian, we completely determine this unit group.

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