Transitive Automorphism Group Research Articles

Every exponential group supports a positive harmonic function

We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, our results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of automorphisms for which the directed balls grow exponentially.

On m-ovoids of finite classical polar spaces with an irreducible transitive automorphism group

On m-ovoids of finite classical polar spaces with an irreducible transitive automorphism group

Doubly transitive, bilinear dimensional dual hyperovals: The Conclusion

In Yoshiara (2009) [17] determined possible groups which can act as a doubly transitive group of automorphisms on a dimensional dual hyperoval. The non-solvable, doubly transitive dual hyperoval were classified in Dempwolff (2018). By Yoshiara (2009) solvable, doubly transitive dual dimensional hyperovals are dual hyperovals defined over F2. If such a hyperoval has rank n, the doubly transitive group has the form G=E⋅H (semidirect product), E is a normal, elementary abelian group of order 2n in G and H≤ΓL(1,2n) acts transitively by conjugation on the non-trivial elements of E. The solvable, doubly transitive, dual hyperovals have been classified by Yoshiara (2008) and the author (Dempwolff 2015) provided the dual hyperoval is bilinear and GL(1,2n)≤H. In particular the doubly transitive, bilinear dual hyperovals are known if (n,2n−1)=1. If (n,2n−1)≠1 the group ΓL(1,2n) contains subgroups H, such that GL(1,2n)⁄≤H and H acts transitively by conjugation on the non-trivial elements of E. Our paper addresses this case and closes the investigation of doubly transitive, bilinear dual hyperovals.

ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

In the present paper, we classify abelian antipodal distance-regular graphs \(\Gamma\) of diameter 3 with the following property: \((*)\) \(\Gamma\) has a transitive group of automorphisms \(\widetilde{G}\) that induces a primitive almost simple permutation group \(\widetilde{G}^{\Sigma}\) on the set \({\Sigma}\) of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank \({\rm rk}(\widetilde{G}^{\Sigma})\) of \(\widetilde{G}^{\Sigma}\) equals 2 moreover, all such graphs are now known. Here we focus on the case \({\rm rk}(\widetilde{G}^{\Sigma})=3\).Under this condition the socle of \(\widetilde{G}^{\Sigma}\) turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs \(\Gamma\) with the property \((*)\) such that \(rk(\widetilde{G}^{\Sigma})=3\) and the socle of \(\widetilde{G}^{\Sigma}\) is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for \(\widetilde{G}^{\Sigma}\). We follow a classification scheme that is based on a reduction to minimal quotients of \(\Gamma\) that inherit the property \((*)\). For each given group \(\widetilde{G}^{\Sigma}\) with simple classical socle of degree \(|{\Sigma}|\le 2500\), we determine potential minimal quotients of \(\Gamma\), applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of \(\Gamma\) in the case of classical socle for \(\widetilde{G}^{\Sigma}\) under condition \(|{\Sigma}|\le 2500.\)

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field $\mathrm{GF}(q)$ with length $q+1$, which are closely related to the action of the projective general linear group of degree two on the projective line. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$-ary BCH codes with length $q+1$, and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of $\mathrm{PGL} (2, p^m)$-invariant codes over $\mathrm{GF} (p^h)$ is completed. As an application of this result, the $p$-ranks of all incidence structures invariant under the projective general linear group $\mathrm{ PGL }(2, p^m)$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a $3$-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms.

An infinite family of antiprimitive cyclic codes supporting Steiner systems $$S(3,8, 7^m+1)$$

Coding theory and combinatorial $t$-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some $t$-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of $3$-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of $3$-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit $3$-transitive automorphism groups.

Doubly transitive lines I: Higman pairs and roux

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian distance-regular antipodal covers of the complete graph.

Closed $\mathrm{G}_2$-structures with a transitive reductive group of automorphisms

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.

Compactness of operators on the Bergman space of the Thullen domain

We study compact operators on the Bergman space of the domain defined by {(z1,z2)∈C2:|z1|2p+|z2|2<1} with p>0 and p≠1. The domain need not be smooth nor have a transitive automorphism group. We give a sufficient condition for the boundedness of various operators on the Bergman space. Under this boundedness condition, we characterize the compactness of operators on the Bergman space of the Thullen domain.

On symmetric intersecting families

We make some progress on a question of Babai from the 1970s, namely: for n,k∈N with k≤n∕2, what is the largest possible cardinality s(n,k) of an intersecting family of k-element subsets of {1,2,…,n} admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k), and show in particular that s(n,k)=o(n−1k−1) as n→∞ if and only if k=n∕2−ω(n)(n∕logn) for some function ω(⋅) that increases without bound, thereby determining the threshold at which ‘symmetric’ intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.

Principal and doubly homogeneous quandles

In the paper we describe the class of principal quandles and we show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among finite simple quandles by several different equivalent properties, such as for instance being doubly homogeneous (i.e. having a doubly transitive automorphism group). A complete description of finite doubly-homogeneous quandles is provided extending the result of Vendramin (J Math Soc Jpn 69(3):1051–1057, 2017) and solving (Bardakov et al. in Monatsh Math 184(4):519–530, 2017, Problem 6.7). We also provide a classification of connected cyclic quandles with several fixed points independently from Lages and Lopes (Quandles of cyclic type with several fixed points. e-Prints arXiv:1803.10487, 2018).

$t$-private information retrieval schemes using transitive codes

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed, and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed–Muller codes serve as an important and explicit class of examples.

On regular 3-wise intersecting families

Ellis and the third author showed, verifying a conjecture of Frankl, that any 3 3 -wise intersecting family of subsets of { 1 , 2 , … , n } \{1,2,\dots ,n\} admitting a transitive automorphism group has cardinality o ( 2 n ) o(2^n) , while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl, and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any 3 3 -wise intersecting family of subsets of { 1 , 2 , … , n } \{1,2,\dots ,n\} that is regular and increasing has cardinality o ( 2 n ) o(2^n) .

A FAMILY OF TWO-DIMENSIONAL LAGUERRE PLANES OF KLEINEWILLINGHÖFER TYPE II.A.2

Kleinewillinghöfer classified Laguerre planes with respect to linearly transitive groups of central automorphisms. Polster and Steinke investigated two-dimensional Laguerre planes and their so-called Kleinewillinghöfer types. For some of the feasible types the existence question remained open. We provide examples of such planes of type II.A.2, which are based on certain two-dimensional Laguerre planes of translation type. With these models only one type, I.A.2, is left for which no two-dimensional Laguerre planes are known yet.

New [formula omitted]-designs and [formula omitted]-designs having [formula omitted] as an automorphism group

In this paper we classify 2-designs and 3-designs with 28 or 36 points admitting a transitive action of the unitary group U(3,3) on points and blocks. We also construct 2-designs and 3-designs with 56 or 63 points and strongly regular graphs on 36, 63 or 126 vertices having U(3,3) as a transitive automorphism group. Further, we show that this completes the classification of 3-designs admitting a transitive action of the group U(3,3), in terms of parameters. A number of the 3-designs and 2-designs obtained in this paper have not been known before up to our best knowledge.

On self-dual double negacirculant codes

Double negacirculant (DN) codes are the analogues in odd characteristic of double circulant codes. Self-dual DN codes are shown to have a transitive automorphism group. Exact counting formulae are derived for DN codes. The special class of length a power of two is studied by means of Dickson polynomials, and is shown to contain families of codes with relative distances satisfying a modified Varshamov–Gilbert bound. This gives an alternative, and effective proof of the result of Chepyzhov, that there are families of quasi-twisted codes above Varshamov–Gilbert.

Big subsets with small boundaries in a graph with a vertex-transitive group of automorphisms

The theory of ends of finitely generated groups and connected locally finite graphs with vertex- transitive groups of automorphisms can be regarded as a theory of Boolean algebras of subsets of or vertex set of with finite boundaries (in the locally finite Cayley graph of or in respectively), considered modulo finite subsets. We develop a more general theory where infinite subsets with finite boundaries are replaced by certain `big' subsets with `small' boundaries.

A classification theorem for boundary 2-transitive automorphism groups of trees

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each vertex contains the alternating group. The outcome of the classification for a fixed tree $T$ is a countable family of groups, all containing two remarkable subgroups: a simple subgroup of index $\leq 8$ and (the semiregular analog of) the universal locally alternating group of Burger-Mozes (with possibly infinite index). We also provide an explicit example showing that the statement of this classification fails for trees of smaller degree.

Cayley properties of merged Johnson graphs

Extending earlier results of Godsil and of Dobson and Malnič on Johnson graphs, we characterise those merged Johnson graphs \(J=J(n,k)_I\) which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order 2. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with vertex-stabilisers of order 2.

Matroid invariants and counting graph homomorphisms

The number of homomorphisms from a finite graph F to the complete graph Kn is the evaluation of the chromatic polynomial of F at n. Suitably scaled, this is the Tutte polynomial evaluation T(F;1−n,0) and an invariant of the cycle matroid of F. De la Harpe and Jaeger [8] asked more generally when is it the case that a graph parameter obtained from counting homomorphisms from F to a fixed graph G depends only on the cycle matroid of F. They showed that this is true when G has a generously transitive automorphism group (examples include Cayley graphs on an abelian group, and Kneser graphs).Using tools from multilinear algebra, we prove the converse statement, thus characterizing finite graphs G for which counting homomorphisms to G yields a matroid invariant. We also extend this result to finite weighted graphs G (where to count homomorphisms from F to G includes such problems as counting nowhere-zero flows of F and evaluating the partition function of an interaction model on F).