2‐(<i>v,k,1</i>) Designs Admitting a Primitive Rank 3 Automorphism Group of Affine Type: The Extraspecial and the Exceptional Classes

Montinaro, Alessandro

doi:10.1002/jcd.21402

Cited by 6 publications

(4 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Devillers classified these linear spaces when the automorphism groups are of almost simple type and grid type [10], [9]. In recent works [1], [16], Biliotti, Francot and Montinaro have completed the classification in the case when the automorphism groups are of affine type.…”

Section: Introductionmentioning

confidence: 98%

Extremely primitive groups and linear spaces

Guan

Zhou

2016

Czech Math J

View full text Add to dashboard Cite

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.

show abstract

Section: Introductionmentioning

confidence: 98%

Extremely primitive groups and linear spaces

Guan

Zhou

2016

Czech Math J

View full text Add to dashboard Cite

show abstract

“…Devillers classified these linear spaces when the automorphism groups are of almost simple type and grid type [5,6]. In [1,14], Biliotti, Francot, and Montinaro have completed the classification in the case where the automorphism groups are of affine type. In these works, the automorphism groups considered are line-transitive, except for the cases of rank 3.…”

Section: Introductionmentioning

confidence: 99%

Alternating groups and point‐primitive linear spaces with number of points being squarefree

Guan

Zhou

2023

J of Combinatorial Designs

View full text Add to dashboard Cite

This paper is a further contribution to the classification of point‐primitive finite regular linear spaces. Let MJX-tex-caligraphicscriptS ${\mathscr{S}}$ be a nontrivial finite regular linear space whose number of points v $v$ is squarefree. We prove that if G≤Aut(S) $G\le \text{Aut}({\mathscr{S}})$ is point‐primitive with an alternating socle, then MJX-tex-caligraphicscriptS ${\mathscr{S}}$ is the projective space PG(3,2) $\text{PG}(3,2)$.

show abstract

“…In particular, for a rank 3 permutation group G on P, if S := (P, L) is a linear space with at least three points on every line such that G Aut(S) and G has exactly two orbits L 1 and L 2 on L, then (P, L 1 ) and (P, L 2 ) are proper partial linear spaces that admit G and have disjoint collinearity relations; in fact, the converse of this statement also holds (see Lemma 2.11). Observe that S is a 2-(v, k, 1) design -that is, a linear space with v points and line-size k -precisely when (P, L 1 ) and (P, L 2 ) both have line-size k. Those 2-(v, k, 1) designs admitting a rank 3 automorphism group G with two orbits on lines have been studied in the special case of affine planes (for example, [3]) and in general when G is an affine primitive group [4,46]. For each partial linear space S in Table 1, we state whether S can be obtained from a 2-(v, k, 1) design using a rank 3 group as above; when this design is an affine plane (that is, when v = k 2 ), we state this instead, and when this affine plane is well known, we give its name.…”

Section: Introductionmentioning

confidence: 99%

Partial linear spaces with a rank 3 affine primitive group of automorphisms

Bamberg

Devillers

Fawcett

et al. 2021

J. London Math. Soc.

View full text Add to dashboard Cite

A partial linear space is a pair (P, L) where P is a non-empty set of points and L is a collection of subsets of P called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms G of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when G is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of AΓL1(q). Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

show abstract

2‐(v,k,1) Designs Admitting a Primitive Rank 3 Automorphism Group of Affine Type: The Extraspecial and the Exceptional Classes

Abstract: 2‐(v,k,1) designs admitting a primitive rank 3 automorphism group G=TG0, where G0 belongs to the Extraspecial Class, or to the Exceptional Class of Liebeck's Theorem in [23], are classified.

Cited by 6 publications

References 25 publications

Extremely primitive groups and linear spaces

Extremely primitive groups and linear spaces

Alternating groups and point‐primitive linear spaces with number of points being squarefree

Partial linear spaces with a rank 3 affine primitive group of automorphisms

Contact Info

Product

Resources

About