Transitive Permutation Groups Research Articles

Prime order derangements in primitive permutation groups

Let G be a transitive permutation group on a finite set Ω of size at least 2. An element of G is a derangement if it has no fixed points on Ω. Let r be a prime divisor of | Ω | . We say that G is r-elusive if it does not contain a derangement of order r, and strongly r-elusive if it does not contain one of r-power order. In this note we determine the r-elusive and strongly r-elusive primitive actions of almost simple groups with socle an alternating or sporadic group.

Block sensitivity of minterm-transitive functions

Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures. In a recent work of this type, Sun (2007) [9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy b s ( f ) = Ω ( N 1 / 3 ) . He also showed that there exists such a function for which b s ( f ) = O ( N 3 / 7 ln N ) . His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) [3]. We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy b s ( f ) = Ω ( N 3 / 7 ) . Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which b s ( f ) = O ( N 3 / 7 ln 1 / 7 N ) .

Vertex-Transitive $q$-Complementary Uniform Hypergraphs

For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b < k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t < k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

Metacirculant tournaments whose order is a product of two distinct primes

In this paper, we prove that non-circulant vertex-transitive tournaments of order p q , where p and q are distinct odd primes, are metacirculant tournaments (defined in Definition 2.1) satisfying some special conditions; see Theorem 1.2. So, in combination with the work in Jing Xu (2010) [11], a complete classification of vertex-transitive p q -tournaments is obtained. As a by-product, we construct examples of non-Cayley vertex-transitive p q -tournaments where q 2 | ( p − 1 ) in Example 2.5. Moreover, applying the classification of vertex-transitive p q -tournaments, we determine all 2-closed (in Wielandt’s sense) odd-order transitive permutation groups of degree p q and show that each of them is the full automorphism group of some tournament.

Finite generation of iterated wreath products

Let (G n , X n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $${\ldots\wr G_2\wr G_1}$$ is topologically finitely generated if and only if the profinite abelian group $${\prod_{n\geq 1} G_n/G'_n}$$ is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power $${G\wr \ldots\wr G\wr G}$$ (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.

Association schemes in which the thin residue is a finite cyclic group

In this article, we investigate association schemes S (on finite sets) in which each element s satisfies s s ⁎ s = { s } . It is shown that these schemes are schurian if the partially ordered set of the intersections of the closed subsets s ⁎ s of S with s ∈ S is distributive. (A scheme is said to be schurian if it arises (in a well-understood way) from a transitive permutation group.) It is also shown that, if these schemes are schurian, the transitive permutation group from which they arise have subnormal one-point stabilizers. As a consequence of the first result one obtains that schemes are schurian if their thin residue is thin and has a distributive normal closed subset lattice (normal subgroup lattice). This implies, for instance, that schemes are schurian if their thin residue is a cyclic group.

Riemann surfaces and restrictively-marked hypermaps

If S is a compact Riemann surface of genus g > 1 then S has at most 84( g − 1) (orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of automorphisms of S and | G | > 24( g − 1) then G is the automorphism group of a regular oriented map (of genus g ) and if | G | > 12(g − 1) then G is the automorphism group of a regular oriented hypermap of genus g (Singerman). We generalise these results and prove that if | G | > g − 1 then G is the automorphism group of a regular restrictedly-marked hypermap of genus g . As a special case we also show that a marked finite transitive permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.

Computation of isomorphisms of coherent configurations

A program computing isomorphisms between association schemes was applied to speed up the computation of normalizers of permutation groups. A transitive permutation group forms an association scheme while a permutation group which may not be transitive forms a coherent configuration. In this paper we discuss the extension of the above program to compute isomorphisms between coherent configurations and show some typical examples of computing normalizers.

On nilpotent subgroups containing non-trivial normal subgroups

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if jG : Ajc expðAÞ, the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by jG : Aj < ExpðGÞ, where ExpðGÞ denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups.

Vertex-transitive tournaments of order a product of two distinct primes

In this paper, we begin a classification of the vertex-transitive tournaments of order pq where p and q are distinct odd primes. In particular we characterize the pq -circulant tournaments; see Theorem 4.3. Moreover, we determine 2-closed (in Wielandt's sense) odd-order transitive permutation groups of degree p and pq by using the classifications of vertex-transitive tournaments of order p and pq . We prove that each such 2-closed group containing a cyclic regular subgroup is the full automorphism group of some circulant tournament.

General form of Humbert's modular equation for curves with real multiplication of $\Delta =5$

We study Humbert's modular equation which characterizes curves of genus two having real multiplication by the quadratic order of discriminant 5. We give it a simple, but general expression as a polynomial in $x_1,\ldots ,x_6$ the coordinate of the Weierstrass points, and show that it is invariant under a transitive permutation group of degree 6 isomorphic to $\frak S_5$. We also prove the rationality of the hypersurface in P5 defined by the generalized modular equation.

A partial generalization of the Livingstone-Wagner Theorem

For a transitive permutation group G on a finite set Ω, the Livingstone–Wagner Theorem states that if G is k -homogeneous then G is ( k − 1)-transitive. It can be conjectured that the number of G -orbits on k -subsets of Ω is greater than or equal to the one on ordered ( k − 1)-tuples of Ω, if |Ω| is sufficiently large. For the simplest case k = 3, we prove this by establishing a result on edge-colorings of complete digraphs.

On Elusive Permutation Groups of Square-Free Degree

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order. In this article we prove that each minimal normal subgroup of an elusive permutation group with square-free degree is abelian. As an almost immediate consequence, it follows that there are no elusive permutation groups of degree 2pq where p, q are distinct odd primes.

Transitive permutation groups of prime-squared degree

We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley graphs of order p-squared.

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups PGL ( 2 , n ) , PSL ( 2 , n ) , PGU ( 3 , n ) , PSU ( 3 , n ) , Sz ( n ) and Ree ( n ) with n a suitable power of a prime number p .

Orbits on n-Tuples

A transitive (infinite) permutation group which has m orbits on ordered pairs of distinct points has at least m n−1 orbits on ordered n-tuples. This is best possible, and groups attaining the bound can be characterised.

A course on Moufang sets

A Moufang set is essentially a doubly transitive permutation group such that the point stabilizer contains a normal subgroup which is regular on the remaining points. These regular normal subgroups are called the root groups and they are assumed to be conjugate and to generate the whole group. Moufang sets play an significant role in the theory of buildings, they provide a tool to study linear algebraic groups of relative rank one, and they have (surprising) connections with other algebraic structures. In these course notes we try to present the current approach to Moufang sets. We include examples, connections with related areas of mathematics and some proofs where we think it is instructive and within the scope of these notes.

A note on transitive permutation groups of degree twice a prime

In this note, we consider transitive permutation groups of degree 2 p , where p is an odd prime, that admit blocks of imprimitivity of size 2 but no blocks of imprimitivity of size p .

Characterizing a family of elusive permutation groups

A finite transitive permutation group is said to be elusive if it has no fixed-point free elements of prime order. In this paper we show that all elusive groups G ¼ N z G1 with N an elementary abelian minimal normal subgroup and G1 cyclic, can be constructed from transitive subgroups of AGLð1; p 2 Þ, for p a Mersenne prime, acting on the set of pð p þ 1Þ lines of the a‰ne plane AGð2; pÞ.

A note on Frobenius groups

We give a new approach to the celebrated theorem of Frobenius (describing the structure of Frobenius groups). We also study the subgroup of a finite transitive permutation group generated by the fixed point free elements (derangements) and its relation to the Frobenius–Wielandt theorem.