Abstract We consider 2-designs which admit a group of automorphisms that is flag-transitive and leaves invariant a chain of nontrivial point-partitions. We build on our recent work on 2-designs which ...are block-transitive but not necessarily flag-transitive. In particular we use the concept of the “array” of a point subset with respect to the chain of point-partitions; the array describes the distribution of the points in the subset among the classes of each partition. We obtain necessary and sufficient conditions on the array in order for the subset to be a block of such a design. By explicit construction we show that for any $$s \ge 2$$ s ≥ 2 , there are infinitely many 2-designs admitting a flag-transitive group that preserves an invariant chain of point-partitions of length s . Moreover an exhaustive computer search, using Magma , seeking designs with $$e_1e_2e_3$$ e 1 e 2 e 3 points (where each $$e_i\le 50$$ e i ≤ 50 ) and a partition chain of length $$s=3$$ s = 3 , produced 57 such flag-transitive designs, among which only three designs arise from our construction—so there is still much to learn.
More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only ...if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.
Arc‐transitive bicirculants Devillers, Alice; Giudici, Michael; Jin, Wei
Journal of the London Mathematical Society,
January 2022, 2022-01-00, Letnik:
105, Številka:
1
Journal Article
Recenzirano
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In this paper, we characterise the family of finite arc‐transitive bicirculants. We show that every finite arc‐transitive bicirculant is a normal r$r$‐cover of an arc‐transitive graph that lies in ...one of eight infinite families or is one of seven sporadic arc‐transitive graphs. Moreover, each of these ‘basic’ graphs is either an arc‐transitive bicirculant or an arc‐transitive circulant, and each graph in the latter case has an arc‐transitive bicirculant normal r$r$‐cover for some integer r$r$.
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.
Block‐transitive designs based on grids Alavi, Seyed Hassan; Daneshkhah, Ashraf; Devillers, Alice ...
The Bulletin of the London Mathematical Society,
April 2023, 2023-04-00, Letnik:
55, Številka:
2
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We study point‐block incidence structures (P,B)$(\mathcal {P},\mathcal {B})$ for which the point set P$\mathcal {P}$ is an m×n$m\times n$ grid. Cameron and the fourth author showed that each block ...B$B$ may be viewed as a subgraph of a complete bipartite graph Km,n$\mathbf {K}_{m,n}$ with bipartite parts (biparts) of sizes m,n$m, n$. In the case where B$\mathcal {B}$ consists of all the subgraphs isomorphic to B$B$, under automorphisms of Km,n$\mathbf {K}_{m,n}$ fixing the two biparts, they obtained necessary and sufficient conditions for (P,B)$(\mathcal {P},\mathcal {B})$ to be a 2‐design, and to be a 3‐design. We first reinterpret these conditions more graph theoretically, and then focus on square grids, and designs admitting the full automorphism group of Km,m$\mathbf {K}_{m,m}$. We find necessary and sufficient conditions, again in terms of graph theoretic parameters, for these incidence structures to be t$t$‐designs, for t=2,3$t=2, 3$, and give infinite families of examples illustrating that block‐transitive, point‐primitive 2‐designs based on grids exist for all values of m$m$, and flag‐transitive, point‐primitive examples occur for all even m$m$. This approach also allows us to construct a small number of block‐transitive 3‐designs based on grids.
On flag-transitive 2-(v,k,2) designs Devillers, Alice; Liang, Hongxue; Praeger, Cheryl E. ...
Journal of combinatorial theory. Series A,
January 2021, 2021-01-00, Letnik:
177
Journal Article
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This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism ...group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n≥3. Alongside this analysis we give a construction for a flag-transitive 2-(v,k−1,k−2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG(n−1,3) as input to this construction yields a G-flag-transitive 2-(v,3,2) design where G has socle PSL(n,3) and v=(3n−1)/2. Apart from these designs, our classification yields exactly one other example, namely the complement of the Fano plane.
The distinguishing number of
G
⩽
Sym
(
Ω
)
is the smallest size of a partition of
Ω
such that only the identity of
G
fixes all the parts of the partition. Extending earlier results of Cameron, ...Neumann, Saxl, and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for
GL
(
2
,
3
)
acting on the eight non-zero vectors of
F
3
2
, which has distinguishing number three.
On flag‐transitive imprimitive 2‐designs Devillers, Alice; Praeger, Cheryl E.
Journal of combinatorial designs,
July 2021, 2021-07-00, 20210701, Letnik:
29, Številka:
8
Journal Article
Recenzirano
Odprti dostop
In 1987, Huw Davies proved that, for a flag‐transitive point‐imprimitive 2‐
(
v
,
k
,
λ
) design, both the block‐size
k and the number
v of points are bounded by functions of
λ, but he did not make ...these bounds explicit. In this paper we derive explicit polynomial functions of
λ bounding
k and
v. For
λ
⩽
4 we obtain a list of “numerically feasible” parameter sets
v
,
k
,
λ together with the number of parts and part‐size of an invariant point‐partition and the size of a nontrivial block‐part intersection. Moreover from these parameter sets we determine all examples with fewer than 100 points. There are exactly 11 such examples, and for one of these designs, a flag‐regular, point‐imprimitive
2
−
(
36
,
8
,
4
) design with automorphism group
S
6, there seems to be no construction previously available in the literature.
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