Sharply 1-transitive subsets of certain permutation groups

In this paper we give a method for constructing sharply 1-transitive permutation sets inside a finite permutation group with certain properties and we apply this method to obtain a family of sharply 1-transitive permutation subsets of the sharply 3-transitive permutation group M(p ^2f ) on PG(1, p ^2f ) for p ^f 1 (mod 4).Work supported by G.N.S.A.G.A. and M.P.I.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

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.     

Each Invertible Sharply d-Transitive Finite Permutation Set with d ≥ 4 is a Group

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S ₄, S ₅, A ₆ and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.     

The number of labeled graphs placeable by a given permutation

Let S be a finite set and σ a permutation on S. The permutation σ* on the set of 2-subsets of S is naturally induced by σ. Suppose G is a graph and V(G), E(G) are the vertex set, the edge set, respectively. Let V(G) = S. If E(G) and σ*(E(G)), the image of E(G) by σ*, have no common element, then G is said to be placeable by σ. This notion is generalized as follows. If any two sets of {E(G), (σ¹)*(E(G)),…,(σ^l−1)* (E(G))} have no common element, then G is said to be I-placeable by σ. In this paper, we count the number of labeled graphs which are I-placeable by a given permutation. At first, we introduce the interspaced Ith Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced Ith Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, we show the number of labeled graphs which are I-placeable by a given permutation is a product of the interspaced Ith Lucas numbers. Finally, using a property of the generalized binomial series, we count the number of labeled graphs of size k which are I-placeable by σ. © 1996 John Wiley & Sons, Inc.     

Orbit-equivalent infinite permutation groups

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not 2-transitive, then G=H.     

The nonexistence of 8-transitive graphs

We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.     

Primitive permutation groups with a regular subgroup

This paper starts the classification of the primitive permutation groups (G,Ω) such that G contains a regular subgroup X. We determine all the triples (G,Ω,X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G,Ω,X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G,Ω,X) with . In order to obtain all these triples, we also study the almost simple groups G with GPΩ_2n+1(q). The case GU_n(q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657–673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.     

Homogeneity in infinite permutation groups

It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:

1. G is not (k + 3)-homogeneous.
2. IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA _k .

     

On the sharply 1-transitive subsets of PGL(2,pm)

If p is an odd prime and R is a sharply 1-transitive subset of PGL(2,p^m) which contains the identity but is not a group, then the subgroup generated by R is either PSL(2,p^m) or PGL(2,p^m).work done within the activity of G.N.S.A.G.A. and supported by the Italian Ministry of Public EducationDedicated to Professor Helmut Karzel on his 60^th birthday     

Permutation polytopes and indecomposable elements in permutation groups

Each group G of n×n permutation matrices has a corresponding permutation polytope, P(G):=conv(G)⊂Rn×n. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t,⌊n/2⌋} is a sharp upper bound on the diameter of the graph of P(G). We also show that P(G) achieves its maximal dimension of ²(n−1) precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83-90] on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.     

O-2-transitive automorphism groups with abelian stabilizer of a point

It is proved that an o-2-transitive group of order automorphisms of a totally ordered set with Abelian stabilizer of a point is the permutation groupF={φ(a, b)‖a, bεP, a>0, (x)φ(a, b)=xa+b forxεP} of a totally ordered fieldP. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 289–293, February, 1999.     

Hexavalent (G,s)-transitive graphs

Let X be a finite simple undirected graph with a subgroup G of the full automorphism group Aut(X). Then X is said to be (G, s)-transitive for a positive integer s, if G is transitive on s-arcs but not on (s + 1)-arcs, and s-transitive if it is (Aut(X), s)-transitive. Let G _v be a stabilizer of a vertex v ∈ V (X) in G. Up to now, the structures of vertex stabilizers G _v of cubic, tetravalent or pentavalent (G, s)-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers G _v of connected hexavalent (G, s)-transitive graphs.     

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

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with n a suitable power of a prime number p.     

Uncoverings-by-bases for base-transitive permutation groups

An uncovering-by-bases for a group G acting on a finite set Ω is a set of bases for G such that any r-subset of Ω is disjoint from at least one base in , where r is a parameter dependent on G. They have applications in the decoding of permutation groups when used as error-correcting codes, and are closely related to covering designs. We give constructions for uncoverings-by-bases for many families of base-transitive group (i.e. groups which act transitively on their irredundant bases), including a general construction which works for any base-transitive group with base size 2, and some more specific constructions for other groups. In particular, those for the groups GL (3,q) and AGL (2,q) make use of the theory of finite fields. We also describe how the concept of uncovering-by-bases can be generalised to matroid theory, with only minor modifications, and give an example of this.     

On 2-transitive 3-nets

A 3-net is said to be 2-transitive if it admits a group of direction-preserving automorphisms fixing one of the transversal lines and acting 2-transitively on its points. We classify the 2-transitive finite 3-nets which do not admit a proper 2-transitive 3-subnet, except, possibly, for a subnet of order 2. The result is then extended under a weaker assumption.To Prof. Walter Benz and Prof. Jakob Joussen on their 60^th birthdaywork done within the activity of GNSAGA of CNR and supported by the Italian Ministry for Research and Technology.     

Orbit equivalence and permutation groups defined by unordered relations

For a set Ω an unordered relation on Ω is a family R of subsets of Ω. If R is such a relation we let G(R)\mathcal{G}(R) be the group of all permutations on Ω that preserve R, that is g belongs to G(R)\mathcal{G}(R) if and only if x∈R implies x ^g ∈R. We are interested in permutation groups which can be represented as G=G(R)G=\mathcal{G}(R) for a suitable unordered relation R on Ω. When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group ≠Alt(Ω) and of degree ≥11 is a relation group. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprimitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation.     

On sharply 2-transitive groups with point stabilizer of exponent 2n⋅3

We describe sharply 2-transitive groups whose point stabilizer is a nilpotent {2,3}-group without elements of order 9 and, more generally, in which the third power of each element belongs to the FC-center. In particular, we will prove that these groups are finite.     

1-transitive cyclic orderings

We give a classification of all the countable 1-transitive cyclic orderings, being those on which the automorphism group acts singly transitively. We also classify all the countable 1-transitive coloured cyclic orderings, where these are countable cyclic orderings in which each point is assigned a member of a set C, thought of as its ‘colour’, and by ‘1-transitivity’ we now mean that the automorphism group acts singly transitively on each set of points coloured by a fixed colour. We conclude by giving constructions of some uncountable cyclic orderings whose automorphism groups enjoy certain special properties.