Orbit coherence in permutation groups

Monatshefte für Mathematik, 2011

If V is a (possibly infinite) set, G a permutation group on V , v ∈ V , and Ω is an orbit of the stabiliser G v , let G Ω v denote the permutation group induced by the action of G v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G v and G Ω v. Among other things, we prove that if G is transitive, G v is finite, and the N-orbital {(v g , u g) | u ∈ Ω, g ∈ N (G)} is strongly connected (when viewed as a digraph on V), then every simple homomorphic image of a subgroup of G v is also a homomorphic image of a subgroup of G Ω v. To demonstrate that the assumption on finiteness of G v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G v is isomorphic to the modular group PSL(2, Z) ∼ = C 2 * C 3 .

Glasgow Mathematical Journal, 1991

Journal of the London Mathematical Society, 1988

The concept of the 2-closure of a finite permutation group was introduced by Wielandt in [15]. We recall the definition: if G is a finite permutation group on a set ft, the 2-closure G <2) of G is the largest subgroup of Sym (Q) containing G which has the same orbits as G in the induced action on Q x Q. Wielandt obtains a number of properties of G <2> in [15], and there are some interesting applications in [12,13,15]. Clearly, if G is primitive of rank r on Q then so is G <2) , and G (2> is the largest subgroup of Sym (Q) containing G with this property. In particular, if G is 2-transitive on Q. then G (2> is the full symmetric group Sym (Q). It is natural to ask about the size of G (2) when G is simply primitive (that is, primitive but not 2-transitive). For nonabelian simple groups of order less than 10 6 this is one of the themes in [7]. In this paper we use the result of [9] to investigate G (2) whenever G is almost simple, that is, L o G < Aut L for some non-abelian simple group L. Our main result shows that if G is almost simple and simply primitive on ft, then G <2) is in general contained in the normalizer of L in Sym (Q). There are, however, some striking examples where this is not so. Apart from six 'sporadic' rank 3 examples which are probably well known, there are two infinite families with unbounded ranks. THEOREM 1. Let G be an almost simple group with L <i G ^ Aut L and L simple. Assume that G is simply primitive on a set ft of size n. Then G (2) is contained in the normalizer of L in Sym (Q) except in the following examples: (1) L = G 2 (q) with q^3,n = q\q z-\)/2 and the socle of G™ is Q 7 (#); (2) L = Cl 7 (q), n = ^3(^4-1)/(2, ?-1) and the socle of G™ is PQ£(tf); (3) six exceptional examples of rank 3 given in Table 1.

Transactions of the American Mathematical Society, 2004

Let G and X be transitive permutation groups on a set Ω such that G is a normal subgroup of X. The overgroup X induces a natural action on the set Orbl(G, Ω) of non-trivial orbitals of G on Ω. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples (G, X, Ω) where X fixes no elements of Orbl(G, Ω); such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples (G, X, Ω, P) where P is a partition of Orbl(G, Ω) such that X is transitive on P; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple (G, X, Ω) in a TOD (G, X, Ω, P) is exceptional; conversely if an exceptional triple (G, X, Ω) is such that X/G is cyclic of prime-power order, then there exists a partition P of Orbl(G, Ω) such that (G, X, Ω, P) is a TOD. This paper characterizes TODs (G, X, Ω, P) such that X Ω is primitive and X/G is cyclic of primepower order. An application is given to the classification of self-complementary vertex-transitive graphs.

Acta Mathematica Hungarica, 2012

We say that the subgroups G1 and G2 of a group G are mutually permutable if G1 permutes with every subgroup of G2 and G2 permutes with every subgroup of G1. Let G = G1G2. .. Gn be the product of its pairwise permutable subgroups G1, G2,. .. , Gn such that the product GiGj is mutually permutable. We investigate the structure of the finite group G if special properties of the factors G1, G2,. .. , Gn are known. Our results improve and extend some results of Asaad and Shaalan [1], Ezquerro and Soler-Escrivà [9] and Asaad and Monakhov [3].

Algebra Universalis, 1997

In an-permutation group (G, Ω), with Ω a chain andΩ its Dedekind completion, the coincidence of two stabilizer subgroups G δ = G λ (δ, λ ∈Ω) yields a map δg → λg (g ∈ G) from δG to λG, and this map commutes with all the elements of G. Roughly speaking, a tying is such a map. We show that the permutations ofΩ which commute with the tyings are exactly those in the closure of G in the full automorphism group A(Ω) with respect to the coarse stabilizer topology. We term this closure the gate completion of G, written G :. We show that each o-primitive component of G : consists of those permutations of the closure of the corresponding G component which respect the orbits of the points which are "tied" to nonsingleton o-blocks. Finally, we show that any two representations of the same lattice-ordered group which are complete and without dead segments give rise to the same G : , and that in this case G : is the α-completion of G.

2002

Various descending chains of subgroups of a finite permutation group can be used to define a sequence of &#39;basic&#39; permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the traditional choice for this purpose, but some combinatorial applications require different kinds of basic groups, such as quasiprimitiv e groups, that are defined by properties of their normal subgroups. Quasiprimitive groups admit similar analyses to primitive groups, share many of their properties, and have been used successfully, for example to study s-arc transitive graphs. Moreover investigating them has led to new results about finite simple groups.

Arxiv preprint math/0611758, 2006

If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is the orbit (a, b)^G is called an orbital digraph of G. A locally finite digraph X has more than one end if there exists a finite set of vertices Y such that the induced digraph X \ Y contains at least two infinite connected components; if there exists such a set containing precisely one element, then X has connectivity one. In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in another paper (Infinite primitive digraphs) by the author.

J. Austral. Math. Soc. Ser. …, 1988

2009

We give a general constructive proof for hierarchical coordinatizations (Lagrange Decompositions) of permutation groups. The generalization originates from the investigation of how the subgroup chains of finite permutation groups yield different coordinate systems. The study is motivated by the practical needs and the verification of an existing computational implementation. Large scale machine calculated examples are also presented.