Flag-Transitive 3- (v, k,3) Designs and PSL(2,q) Groups

2021 ◽  
Vol 28 (01) ◽  
pp. 33-38
Author(s):  
Shaojun Dai ◽  
Shangzhao Li
Keyword(s):  
Linear Group ◽  
Automorphism Groups ◽  
Two Dimensional ◽  
Text Design ◽  
Projective Linear Group

This article is a contribution to the study of the automorphism groups of 3-[Formula: see text] designs. Let [Formula: see text] be a non-trivial 3-[Formula: see text] design. If a two-dimensional projective linear group [Formula: see text] acts flag-transitively on [Formula: see text], then [Formula: see text] is a 3-[Formula: see text] or 3-[Formula: see text] design.

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

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.

Homomorphisms of two-dimensional linear groups over Laurent polynomial rings and Gaussian domains

1991 ◽  
Vol 109 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Yu Chen
Keyword(s):  
Commutative Ring ◽  
Linear Group ◽  
Unit Group ◽  
Commutative Rings ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Linear Groups ◽  
Two Dimensional ◽  
Euclidean Domain ◽  
Invertible Matrices

Let GL2(R) be the general linear group of 2 × 2 invertible matrices in M2(R) over a commutative ring R with 1 and SL2(R) be the special linear group consisting of 2 × 2 matrices over R with determinant 1. In this paper we determine the homomorphisms from GL2 and SL2, as well as their projective groups, over Laurent polynomial rings to those groups over Gaussian domains, i.e. unique factorization domains (cf. Theorems 1, 2, 3 below). We also consider more generally the homomorphisms of non-projective groups over commutative rings containing a field which are generated by their units (cf. Theorems 4 and 5). So far the homomorphisms of two-dimensional linear groups over commutative rings have only been studied in some specific cases. Landin and Reiner[7] obtained the automorphisms of GL2(R), where R is a Euclidean domain generated by its units. When R is a type of generalized Euclidean domain with a degree function and with units of R and 0 forming a field, Cohn[3] described the automorphisms of GL2(R). Later, Cohn[4] applied his methods to the case of certain rings of quadratic integers. Dull[6] has considered the automorphisms of GL2(R) and SL2(R), along with their projective groups, provided that R is a GE-ring and 2 is a unit in R. McDonald [9] examined the automorphisms of GL2(R) when R has a large unit group. The most recent work of which we are aware is that of Li and Ren[8] where the automorphisms of E2(R) and GE2(R) were determined for any commutative ring R in which 2, 3 and 5 are units.

Automorphism groups and invariant theory on ℙN

2018 ◽  
Vol 17 (09) ◽  
pp. 1850162 ◽  
Author(s):  
João Alberto de Faria ◽  
Benjamin Hutz
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Invariant Theory ◽  
Automorphism Groups ◽  
Finite Subgroup ◽  
Fixed Point Algorithm ◽  
Projective Linear Group ◽  
Rationality Problems ◽  
Stabilizer Group ◽  
A Finite Group

Let [Formula: see text] be a field and [Formula: see text] a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group [Formula: see text]. The group of automorphisms, or stabilizer group, of a given [Formula: see text] for this action is known to be a finite group. In this paper, we apply methods of invariant theory to automorphism groups by addressing two mainly computational problems. First, given a finite subgroup of [Formula: see text], determine endomorphisms of [Formula: see text] with that group as a subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism, determine its automorphism group. In particular, we extend the Faber–Manes–Viray fixed-point algorithm for [Formula: see text] to endomorphisms of [Formula: see text]. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.

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