scholarly journals On plane curves given by separated polynomials and their automorphisms

2020 ◽  
Vol 20 (1) ◽  
pp. 61-70
Author(s):  
Matteo Bonini ◽  
Maria Montanucci ◽  
Giovanni Zini
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
Plane Curve ◽  
Plane Curves ◽  
Prime Field ◽  
Full Automorphism Group ◽  
Ag Codes ◽  
Finite Prime Field

AbstractLet 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + … + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms.

scholarly journals Equality of certain automorphism groups of finite p-groups

2016 ◽  
Vol 15 (03) ◽  
pp. 1650056
Author(s):  
Deepak Gumber ◽  
Hemant Kalra
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Recent Past ◽  
Full Automorphism Group ◽  
Technical Lemma ◽  
Necessary And Sufficient

Let [Formula: see text] be a finite [Formula: see text]-group and let Aut([Formula: see text]) denote the full automorphism group of [Formula: see text]. In the recent past, there has been interest in finding necessary and sufficient conditions on [Formula: see text] such that certain subgroups of Aut([Formula: see text]) are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.

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 On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

scholarly journals A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2935
Author(s):  
Bo Ling ◽  
Wanting Li ◽  
Bengong Lou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Automorphism Groups ◽  
Cayley Graphs ◽  
Vital Role ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Base Group

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Jordan property and automorphism groups of normal compact Kähler varieties

2018 ◽  
Vol 20 (03) ◽  
pp. 1750024 ◽  
Author(s):  
Jin Hong Kim
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Projective Varieties ◽  
Full Automorphism Group ◽  
Identity Component ◽  
Arbitrary Dimensions ◽  
Type Decomposition

It has been recently shown by Meng and Zhang that the full automorphism group [Formula: see text] is a Jordan group for all projective varieties in arbitrary dimensions. The aim of this paper is to show that the full automorphism group [Formula: see text] is, in fact, a Jordan group even for all normal compact Kähler varieties in arbitrary dimensions. The meromorphic structure of the identity component of the automorphism group and its Rosenlicht-type decomposition play crucial roles in the proof.

scholarly journals A few remarks on invariable generation in infinite groups

2020 ◽  
pp. 1-22
Author(s):  
Gil Goffer ◽  
Gennady A. Noskov
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Lie Group ◽  
Automorphism Groups ◽  
Transcendence Degree ◽  
Linear Groups ◽  
Infinite Groups ◽  
Prime Field ◽  
Regular Tree

A subset [Formula: see text] of a group [Formula: see text] invariably generates [Formula: see text] if [Formula: see text] is generated by [Formula: see text] for any choice of [Formula: see text]. A topological group [Formula: see text] is said to be [Formula: see text] if it is invariably generated by some subset [Formula: see text], and [Formula: see text] if it is topologically invariably generated by some subset [Formula: see text]. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group [Formula: see text] and the automorphism group of a regular tree are [Formula: see text], and that the groups [Formula: see text] are not [Formula: see text] for countable fields of infinite transcendence degree over a prime field.

scholarly journals Automorphism groups of Cayley digraphs of ${\Bbb Z}_p^3$

10.37236/238 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Edward Dobson ◽  
István Kovács
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Full Automorphism Group ◽  
Product Action

We calculate the full automorphism group of Cayley digraphs of ${\Bbb Z}_p^3$, $p$ an odd prime, as well as determine the $2$-closed subgroups of $S_m \wr S_p$ with the product action.

Symmetric designs admitting flag-transitive and point-primitive almost simple automorphism groups of Lie type

2017 ◽  
Vol 16 (10) ◽  
pp. 1750192
Author(s):  
Yajie Wang ◽  
Shenglin Zhou
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Symmetric Design ◽  
Symmetric Designs ◽  
Full Automorphism Group ◽  
Design Formula ◽  
Groups Of Lie Type

Let [Formula: see text] be a subgroup of the full automorphism group of a [Formula: see text]-[Formula: see text] symmetric design [Formula: see text]. If [Formula: see text] is flag-transitive and point-primitive, then Soc[Formula: see text] cannot be [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals Automorphisms and Enumeration of Switching Classes of Tournaments.

10.37236/1516 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
L. Babai ◽  
P. J. Cameron
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Probability Distributions ◽  
Explicit Construction ◽  
Linear Orders ◽  
Full Automorphism Group ◽  
Odd Order ◽  
Vertex Set ◽  
Countably Infinite ◽  
Switching Class

Two tournaments $T_1$ and $T_2$ on the same vertex set $X$ are said to be switching equivalent if $X$ has a subset $Y$ such that $T_2$ arises from $T_1$ by switching all arcs between $Y$ and its complement $X\setminus Y$. The main result of this paper is a characterisation of the abstract finite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Moreover, if $G$ is such a group, then there is a switching class $C$, with Aut$(C)\cong G$, such that every subgroup of $G$ of odd order is the full automorphism group of some tournament in $C$. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of random $G$-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group $G$, acting on a set $X$, is contained in the automorphism group of some switching class of tournaments with vertex set $X$ if and only if the Sylow 2-subgroups of $G$ are cyclic or dihedral and act semiregularly on $X$. Applying this result to individual permutations leads to an enumeration of switching classes, of switching classes admitting odd permutations, and of tournaments in a switching class. We conclude by remarking that both the class of switching classes of finite tournaments, and the class of "local orders" (that is, tournaments switching-equivalent to linear orders), give rise to countably infinite structures with interesting automorphism groups (by a theorem of Fraïssé).

On the Determination of Plane Curves by means of Assigned Singularities

1931 ◽  
Vol 27 (3) ◽  
pp. 291-305 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Elliptic Curve ◽  
Sufficient Conditions ◽  
Plane Curve ◽  
Plane Curves ◽  
Multiple Points ◽  
Pass Through

The principal object of the present paper is to give a set of sufficient conditions, of considerable generality, under which a plane curve assigned to pass through certain points with given multiplicities shall be irreducible. The case considered is that in which the assigned multiple points of the curve are of such generality that the conditions presented to the curve at these points are independent. We shall shew that subject to certain explicit restrictions on the points, which ensure, for example, that no three points shall be collinear whose prescribed multiplicities in aggregate exceed the order of the curve, the curve determined by the points will be irreducible if its freedom is not negative, save in the case when it degenerates into a repeated elliptic curve. A familiar example of this last case arises in the problem of constructing a sextic with nine assigned nodes; in general the curve in question consists of the unique cubic through the nine points, counted twice.

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