Transitive Group Research Articles

Construction of Finite 2-Transitive Groups Based on the Concept of Near Field

Transitive permutation groups were studied basically based on the idea of a near field. The article mainly considered Dickson Near field to construct groups which are 2-transitive. To achieve the result, some restrictions were observed in relation to the field. The research went through the construction with the help of the classification scheme for the finite primitive group proposing that the subgroup of the group is the stabilizer of the group. The idea of this construction has relevance in coding theory.

Finitely generated simple sharply 2-transitive groups

We construct the first examples of infinite sharply 2-transitive groups which are finitely generated. Moreover, we construct such a group that has Kazhdan property (T), is simple, has exactly four conjugacy classes and we show that this number is as small as possible.

On 2-distance primitive graphs of prime valency

A vertex transitive non-complete graph is called 2-distance primitive if the stabilizer of a vertex is primitive on both the first and the second step neighbourhoods. In 2019, Jin, Huang and Liu proved that all connected 2-distance primitive graphs of prime valency must belong to some known families of distance-transitive graphs if the socle of the stabilizer of a vertex is not a 2-transitive linear group with some further restrictions. They posed a problem to determine the connected 2-distance primitive graphs of prime valency when the socle of the stabilizer of a vertex is a 2-transitive linear group. In this paper, we make a crucial progress to this problem by proving that either the graph under consideration in this problem is the folded 5-cube or the socle of the stabilizer of a vertex is PSL(2,q) and the graph has girth 4 and contains no 5-cycles. We also give a short proof of the main result of Jin, Huang and Liu (2019), and we find some new graphs that are not in the list of Jin, Huang and Liu (2019).

Rank three innately transitive permutation groups and related 2-transitive groups

Rank three innately transitive permutation groups and related 2-transitive groups

Simple sharply 2-transitive groups

We construct infinite simple sharply 2-transitive groups. Our result answers an open question of Peter Neumann. In fact, we prove that every sharply 2-transitive groupGGof characteristic 0 embeds into a simple sharply 2-transitive group.

Covers of complete graphs and related association schemes

We investigate the association schemes Inv(G) that are formed by the collection of orbitals of a permutation group G, for which the (underlying) graph Γ of a basis relation is a distance-regular antipodal cover of the complete graph. The group G can be regarded as an edge-transitive group of automorphisms of Γ and induces a 2-homogeneous permutation group on the set of its antipodal classes, which is either almost simple and 2-transitive, or affine. Using the classification of the finite 2-transitive permutation groups, we determine every possibility for Γ except for some special cases. This allows us to obtain a classification of such schemes Inv(G) provided that G is quasi-simple. We also give a general characterization of edge-transitive distance-regular antipodal covers of complete graphs in the almost simple case in terms of graphs of basis relations of schemes Inv(G) with quasi-simple group G. Then, we find several constructions of Deza graphs that are associated with such a scheme Inv(G). Finally, we establish isomorphisms between some graphs of basis relations of Inv(G) with G≃SU3(q) and abelian covers related to generalised quadrangles.

Circulant association schemes on triples

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {\em association scheme on triples} (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called {\em circulant} ASTs and the corresponding ternary relations are called {\em circulant relations}. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a {\em thin circulant}, plays a key role in describing the structure of circulant ASTs. We outline several open questions. 

Orbits of Sylow subgroups of finite permutation groups

We say that a finite group G acting on a set Ω has Property(⁎)p for a prime p if Pω is a Sylow p-subgroup of Gω for all ω∈Ω and Sylow p-subgroups P of G. Property (⁎)p arose in the recent work of Tornier (2018) on local Sylow p-subgroups of Burger–Mozes groups, and he determined the values of p for which the alternating group An and symmetric group Sn acting on n points has Property (⁎)p. In this paper, we extend this result to finite 2-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property (⁎)p for an allowable prime p.

Non-split linear sharply 2-transitive groups

We give examples of countable linear groups $\Gamma < \operatorname {SL}_{{\mathbf {3}}}({\mathbf {R}})$, with no nontrivial normal abelian subgroups, that admit a faithful sharply $2$-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent in [J. Eur. Math. Soc. 19 (2017), pp. 2895–2910]. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.

All 2-transitive groups have the EKR-module property

We prove that every 2-transitive group has a property called the EKR-module property. This property gives a characterization of the maximum intersecting sets of permutations in the group. Specifically, the characteristic vector of any maximum intersecting set in a 2-transitive group is a linear combination of the characteristic vectors of the stabilizers of points and their cosets. We also consider when the derangement graph of a 2-transitive group is connected and when a maximum intersecting set is a subgroup or a coset of a subgroup.

Generalized right near domains and sharply 2-transitive groups

In this paper, we study the generalized right near domains and its relations with sharply [Formula: see text]-transitive groups. We find a necessary and sufficient condition when a generalized right near domain is a field. Also, we have found the isomorphism classes of right near domains in a sharply [Formula: see text]-transitive group.

Hematological and biochemical profiles of Mangalarga Marchador mares in the transition period bred on pasture

ABSTRACT The present study aimed to evaluate the effects of the transition period on hematological and biochemical constituents in Mangalarga Marchador mares. Forty-eight mares were used to form a maintenance group (MG) and transition group (TG), formed by pregnant mares and, after delivery, infants. Blood samples were collected at the following times: T-60 (60 d pre-delivery), T-30 (30 d pre-delivery), T-15 (15 d pre-delivery), T0 (first 6h post-delivery), T15 (15 d post-delivery), T30 (30 d post-delivery), and T60 (60 d post-delivery). The TG had lower values (P&lt; 0.05) of red blood cells, hematocrit and hemoglobin at T0, T15, T30 and T60 times than MG. The mean corpuscular volume was lower in MG (P&lt; 0.05) than in TG (T0, T15, T30 and T60) and mean corpuscular hemoglobin concentration was higher (P&lt; 0.05) in MG than in TG (T15, T30 and T60). On the other hand, the diameter distribution of red blood cells presented a lower value (P&lt; 0.05) in MG than in TG (T15 and T30). Mares in transition period presented regenerative anemia. The results demonstrate physiological metabolic variations of different intensities during pregnancy, delivery and early lactation.

Minimal binary 2-neighbour-transitive codes

The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F2 for finite 2-transitive permutation groups.

Transition period produces changes in blood and body composition in mares

ABSTRACT: Clinical and metabolic evaluation is necessary for the monitoring of pregnant and lactating mares, as they reflect the health of the animal. The body condition of the mare is an indicator of reproductive efficiency. The study aimed to determine the possible variations in body and metabolic condition in Mangalarga Marchador mares during the transition period. Forty-eight mares distributed in two groups were used: Maintenance Group (MG), composed of non-pregnant and non-lactating mares, and Transition Group (TG), formed by pregnant mares and who after delivery became lactating. Analyzes were performed in the times T-60, T-30 and T-15 before delivery, first six hours (T0) after delivery and T15, T30 and T60 days after delivery. MG was evaluated only at one time (T-60). Body weight and fat-free mass differed (P&lt;0.05) between the groups. The percentage of fat was lower in MG. Mares had a higher fat percentage in TG at T-60 and T-30 times. There was a difference (P&lt;0.05) in the amount of cholesterol between MG and TG (T0, T15 and T30). Triglycerides were different between the groups. TG showed higher concentrations of non-esterified fatty acids (P&lt;0.05). There was a higher amount of glucose in TG (delivery and lactation) when compared to mares in MG. Changes in body condition and metabolic constituents occurred in the animals resulting from physiological adaptations of the transition period. The energetic components are the most affected from the transition period, with intense fat mobilization to supply the body demands.

The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

A linear group G ≤ G L ( V ) G\le GL(V) , where V V is a finite vector space, is called 1 2 \frac {1}{2} -transitive if all the G G -orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 \frac {1}{2} -transitive linear groups. As a consequence we complete the determination of the finite 3 2 \frac {3}{2} -transitive permutation groups – the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the ( k + 1 2 ) (k+\frac {1}{2}) -transitive groups for integers k ≥ 2 k\ge 2 .

Biochemical profile of Mangalarga Marchador mares during the last third of pregnancy, foaling and lactation

ABSTRACT: Adaptive changes in metabolism may occur under specific physiological conditions, being necessary the knowledge of the biochemical constituents that can undergo variations during the sixty days before foaling, until the first sixty days of lactation. The present study aimed to identify and evaluate the biochemical profile of Mangalarga Marchador mares during the last third of pregnancy, foaling and lactation. For this, 48 mares were used, which formed a maintenance group (MG), composed of non-pregnant mares and an experimental group, called the transition group (TG), formed by pregnant mares. Blood samples were collected at the following times: MG was evaluated in a single time (beginning of all collections); TG was evaluated in seven times [T-60, T-30, T-15 (days before foaling), T0 (first six hours after foaling), T15, T30, T60 (days after foaling)]. All animals were submitted to biochemical analysis, in which the concentrations of total calcium, chloride, phosphorus, magnesium, sodium, potassium, uric acid, creatinine, and urea were measured. The two groups differed significantly (P&lt;0.05) in calcium, magnesium, sodium, potassium, creatinine, uric acid and urea concentration. Variation between times was also observed for these compounds. However, chloride and phosphorus did not differ between MG and TG. It was concluded that the last third of pregnancy, foaling and lactation exerts major influence on the biochemical constituents of blood through adjustments to maintain homeostasis in mares.

The covering radii of the 2-transitive unitary, Suzuki, and Ree groups

Abstract We study the covering radii of 2-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.

Flag-transitive 4-(v, k, 3) designs and PSL(2, q) groups

Among the properties of homogeneity of incidence structures, flag-transitivity obviously is a particularly important and natural one. Originally, Buekenhout et al. reached a classification of flag-transitive Steiner 2-designs. Recently, Huber completely classified all flag-transitive Steiner t-designs with t ≤ 6 using the classification of the finite 2-transitive permutation groups. Hence the determination of all flag-transitive t-designs with λ ≥ 2 has remained of particular interest and has been known as a long-standing and still open problem.This article is a contribution to the study of the automorphism groups of 4-(v, k, 3) designs. Let S=(P,B) be a non-trivial 4-(q+1,k,3) design. If PSL(2, q) acts flag-transitively on S, then S is a 4-(168,12,3) design and GB is conjugate to A4 or Z12.

A sharply 2-transitive group without a non-trivial abelian normal subgroup

We show that any group G is contained in some sharply 2-transitive group \mathcal G without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups \mathcal G that we construct have no fixed points.

On sharply 2-transitive groups with generalized finite elements

We prove Jordan’s Theorem for infinite sharply 2-transitive groups satisfying the finiteness (a, b)-condition, with |a| · |b| even.