Flag-transitive Automorphism Group Research Articles

Flag-transitive automorphism groups of 2-designs with λ ≥ (r,λ)2 are not product type

In this note we show that a flag-transitive automorphism group G of a non-trivial 2-(v,k,λ) design with λ≥(r,λ)2 is not of product action type. In conclusion, G is a primitive group of affine or almost simple type.

Finite simple groups on triple systems

Let D be a triple system, and let G be a finite simple group. In this paper we almost determine all possibilities of D admitting G as its flag-transitive automorphism group.

Reduction for flag-transitive symmetric designs with k > λ(λ − 2)

Let G be a flag-transitive automorphism group of a symmetric (v,k,λ) design D with k>λ(λ−2). O'Reilly Regueiro proved that if G is point-imprimitive, then D has parameters (v,k,λ)=(λ2(λ+2),λ(λ+1),λ). In the present paper, we consider the case that G is point-primitive. By applying the O'Nan-Scott Theorem, we prove that G must be of affine type or almost simple type.

On 2-(v,k,λ) Designs with Flag-Transitive Automorphism Groups of Product Action Type

In this paper we present two new families of 2-[Formula: see text] designs with a flag-transitive and point-primitive automorphism group of product action type. More surprisingly, one of them is still a family of 2-[Formula: see text] designs with a flag-transitive and point-imprimitive automorphism group.

A classification of flag-transitive 2-(k2,k,λ) designs with λ|k

The pairs (D,G), where D is a non-trivial 2-(k2,k,λ) design with λ|k and G is a flag-transitive automorphism group of D are classified except for k a power of a prime and G⩽AΓL1(k2).

Almost simple groups as flag-transitive automorphism groups of symmetric designs with λ prime

In this article, we study symmetric designs with λ prime admitting a flag-transitive and point-primitive automorphism groups G of almost simple type. We prove that either D is one of the six well-known examples of biplanes and triplanes, or D is the point- hyperplane design of PG(n−1,q) with λ = (q^{n−2}−1)/(q−1) prime and X = PSLn(q).

A classification of 2-designs with prime replication numbers and flag-transitive automorphism groups

A classification of 2-designs with prime replication numbers and flag-transitive automorphism groups

Flag-transitive 2-(v,k,λ) designs with λ prime and alternating socle

This paper is a contribution to the classification of 2-designs admitting a flag-transitive automorphism group. Let [Formula: see text] be a 2-[Formula: see text] design with [Formula: see text] prime and [Formula: see text] be a flag-transitive, point-primitive automorphism group of [Formula: see text]. We prove that if the socle of [Formula: see text] is an alternating group, then [Formula: see text] has parameters [Formula: see text] or [Formula: see text].

On flag-transitive automorphism groups of 2-designs

In this paper, we study flag-transitive 2-designs. Let D be a 2-(v,k,λ) design with (v−1,k−1)≤(v−1)12 admitting a flag-transitive automorphism group G. We prove that G is of affine, almost simple type, or product type with rank(G)=3. In particular, if (v−1,k−1)=3 or 4, then G is of affine, almost simple type except the case that Soc(G)=A5×A5 and D is a 2-(25,4,12) or 2-(25,4,18) design. Finally, an application to the case where k is prime is also discussed.

Flag-transitive automorphism groups of 2-designs with λ ≥ (r,λ)2 and k odd

This paper aims to prove that flag-transitive automorphism groups of [Formula: see text]-[Formula: see text] designs with [Formula: see text] and [Formula: see text] odd are not of product action type.

Block Designs with $$\gcd (r,\lambda )=1$$ Admitting Flag-Transitive Automorphism Groups

In this paper, we present a classification of $$\mathbf{2}$$ -designs with $$\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}$$ admitting flag-transitive automorphism groups. If $${{\varvec{G}}}$$ is a flag-transitive automorphism group of a non-trivial $$\mathbf{2}$$ -design $$\mathcal {{\varvec{D}}}$$ with $$\mathbf{gcd}({{\varvec{r}}},{\varvec{\lambda }})=\mathbf{1}$$ , then either $$(\mathcal {{\varvec{D}}},{{\varvec{G}}})$$ is one of the known examples described in this paper, or $$\mathcal {{\varvec{D}}}$$ has $${{\varvec{q}}} = {{\varvec{p}}}^{{{\varvec{d}}}}$$ points with $${{\varvec{p}}}$$ prime and $${{\varvec{G}}}$$ is a subgroup of $$\mathbf{A}{\varvec{\Gamma }}{} \mathbf{L}_{\mathbf{1}}({{\varvec{q}}})$$ .

Groups \(S_n \times S_m\) in construction of flag-transitive block designs

In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

Classification of the non-trivial 2-$$(v,k,\lambda )$$ designs, with $$ (r,\lambda )=1$$ and $$\lambda >1$$, admitting a non-solvable flag-transitive automorphism group of affine type

Classification of the non-trivial 2-$$(v,k,\lambda )$$ designs, with $$ (r,\lambda )=1$$ and $$\lambda >1$$, admitting a non-solvable flag-transitive automorphism group of affine type

Alternating groups and flag-transitive non-symmetric 2-(v,k,λ) designs with λ ≥ (r,λ)2

In this paper, all non-symmetric 2-(v,k,λ) designs with λ≥(r,λ)2 are classified which admit a flag-transitive group of automorphisms with an alternating socle. Up to isomorphism, our classification shows exactly 20 such designs, and, for each of these designs, we explicitly determine the corresponding flag-transitive automorphism groups.

A classification of flag-transitive block designs

In this article, we investigate 2- $$(v,k,\lambda )$$ designs with $$\gcd (r,\lambda )=1$$ admitting flag-transitive automorphism groups G. We prove that if G is an almost simple group, then such a design belongs to one of the eight infinite families of 2-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $${\mathcal {D}}$$ is a symmetric $$(v,k,\lambda )$$ design with $$\gcd (k,\lambda )=1$$ admitting a flag-transitive automorphism group G, then either $$G\leqslant \mathrm {A}\Gamma \mathrm {L}_{1}(q)$$ for some odd prime power q, or $${\mathcal {D}}$$ is the point-hyperplane design or the unique Hadamard design with parameters (11, 5, 2).

Almost simple groups with socle PSp4(q) as flag-transitive automorphism groups of symmetric designs

In this article, we study symmetric designs admitting a flag-transitive and point-primitive automorphism group whose socle is a symplectic group of dimension four. We prove a reduction theorem severely restricting the possible parameters of such designs, and we show that such a parameter set belongs to one of the eight infinite families of parameter sets. For four families of these parameter sets, we know two sporadic examples of symmetric designs with parameters (45,12,3) and (36,15,6) arose from PSp4(3) and two infinite families of 2-designs namely the complement of a point-hyperplane design of PG(3,q) and the complement of a Higman design. The existence of any symmetric designs with the remaining four parameter sets is still open.

Reduction of flag-transitive automorphism groups of 2-(v,k,λ) designs with (r − λ,k)=1

The paper studies flag-transitive groups of automorphisms of 2-(v,k,λ) designs D with (r−λ,k)=1. In 1968, Peter Dembowski proved that such groups of automorphisms act primitively on the set of points of D. The purpose of the present paper is to reduce the structure of these groups even more, we show that they are affine or almost simple.

Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.

Flag-transitive 2-(υ, 5, λ) designs with sporadic socle

We state that the flag-transitive automorphism group of a 2-(υ, 5, λ) design $${\mathscr{D}}$$ is primitive of affine type or almost simple type. We also find that there are up to isomorphism 20 2-(υ, 5, λ) designs admitting flag-transitive automorphism groups with socle of a sporadic simple group.

A note on flag-transitive automorphism groups of 2-designs with $$\lambda \ge (r,\lambda )^2$$

In this paper, we prove that flag-transitive automorphism groups of 2-designs with $$10^3\ge \lambda \ge (r,\lambda )^2$$ are point-primitive of affine or almost simple type.