Finite Permutation Groups Research Articles

Construction of the irreducible modular representations of a finite group

A complete procedure is described for constructing the irreducible KG-modules and their Brauer characters, where K is a finite field of characteristic p and G is a finite permutation or matrix group. The central idea is to construct a sequence {S1,…,Sn} of KG-modules, each having relatively small dimension, such that each Si has one or more irreducible constituents that are not constituents of S1,…,Si−1. The Meataxe, used in conjunction with condensation, is used to extract the new irreducibles from each Si. The algorithm has been implemented in Magma and is capable of constructing irreducibles of dimension over 200000.

Subgroups of minimal index in polynomial time

By applying an old result of Y. Berkovich, we provide a polynomial-time algorithm for computing the minimal possible index of a proper subgroup of a finite permutation group [Formula: see text]. Moreover, we find that subgroup explicitly and within the same time if [Formula: see text] is given by a Cayley table. As a corollary, we get an algorithm for testing whether or not a finite permutation group acts on a tree non-trivially.

Bounding the composition length of primitive permutation groups and completely reducible linear groups

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.

Approximate transitivity of the ergodic action of the group of finite permutations of on

In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

Primitive Genus Two Groups of Meta Type

This paper is a contribution to the classification of the finite primitive permutation groups of genus two. We consider the case of affine groups. Our main result, Lemma 3.10 gives a complete classification of genus two systems when . We achieve this classification with the aid of the computer algebra system GAP.

The Cycle Polynomial of a Permutation Group

The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles:\[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbital chromatic polynomial of $\Gamma$ and $G$, where $\Gamma$ is a $G$-invariant graph, introduced by the first author, Jackson and Rudd. We pose the general problem of finding all such reciprocal pairs, and give a number of examples and characterisations: the latter include the cases where $\Gamma$ is a complete or null graph or a tree. The paper concludes with some comments on other polynomials associated with a permutation group.

Equivalence of Lattice Orbit Polytopes

Let $G$ be a finite permutation group acting on $\mathbb{R}^d$ by permuting coordinates. A core point (for $G$) is an integral vector $z\in \mathbb{Z}^d$ such that the convex hull of the orbit $Gz$ contains no other integral vectors but those in the orbit $Gz$. Herr, Rehn and Sch\"urmann considered the question for which groups there are infinitely many core points up to translation equivalence, that is, up to translation by vectors fixed by the group. In the present paper, we propose a coarser equivalence relation for core points called normalizer equivalence. These equivalence classes often contain infinitely many vectors up to translation, for example when the group admits an irrational invariant subspace or an invariant irreducible subspace occurring with multiplicity greater than $1$. We also show that the number of core points up to normalizer equivalence is finite if $G$ is a so-called QI-group. These groups include all transitive permutation groups of prime degree. We give an example to show how the concept of normalizer equivalence can be used to simplify integer convex optimization problems.

Finite permutation groups containing a transitive abelian subgroup

ABSTRACTIn this paper, we classify finite permutation groups with a transitive abelian subgroup that are almost simple, quasiprimitive and innately transitive, which extend the results of Li and Praeger that is on finite permutation groups with a transitive cyclic subgroup.

Embedding properties of hereditarily just infinite profinite wreath products

We study infinitely iterated wreath products of finite permutation groups w.r.t. product actions. In particular, we prove that, for every non-empty class of finite simple groups X, there exists a finitely generated hereditarily just infinite profinite group W with composition factors in X such that any countably based profinite group with composition factors in X can be embedded into W. Additionally we investigate when infinitely iterated wreath products of finite simple groups w.r.t. product actions are co-Hopfian or non-co-Hopfian.

UNDECIDABILITY AND THE DEVELOPABILITY OF PERMUTOIDS AND RIGID PSEUDOGROUPS

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a permutoid based on a finite set can be completed to a finite permutation group. In this note we prove Cameron’s conjecture by relating it to our recent work on the profinite triviality problem for finitely presented groups. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable. In an appendix, Steinberg recasts these results in terms of inverse semigroups.

Unique maximal rings of functions

ABSTRACTFor several classes of groups G, we characterize when the near-ring M0(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M0(G) are commutative.

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N≀H = N|Ω| ⋊ H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.

A reduction theorem for primitive binary permutation groups

A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.

A linear-time algorithm for the orbit problem over cyclic groups

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same orbit with respect to a given finite permutation group (represented by their generators) acting on this set of configurations by permuting indices. It is known that the problem is in general as hard as the graph isomorphism problem, whose precise complexity (whether it is solvable in polynomial-time) is a long-standing open problem. In this paper, we consider the restriction of the orbit problem when the permutation group is cyclic (i.e. generated by a single permutation), an important restriction of the problem. It is known that this subproblem is solvable in polynomial-time. Our main result is a linear-time algorithm for this subproblem.

A criterion for a finite permutation group to be transitive

Abstract Let G be a finite permutation group on a finite set Ω. We say that G is quasi-transitive if every 2-point stabiliser has the same order. This notion was introduced by Alan Camina. In particular, Camina established conditions for a quasi-transitive group to be transitive. The aim of this note is to validate Camina's conjecture: A quasi-transitive group G on a finite set Ω is transitive on Ω.

On the relational complexity of a finite permutation group

The relational complexity $$\rho (X,G)$$?(X,G) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or (preferably) precise values for $$\rho $$? in natural cases; and (b) a rough determination of the primitive permutation groups with $$\rho $$? either very small (bounded) or very large (much larger than the logarithm of the degree). The rough version of (a) is relevant to (b). Our main result is an explicit characterization of the binary ($$\rho =2$$?=2) primitive affine permutation groups. We also compute the precise relational complexity of $${{\mathrm{Alt}}}_n$$Altn acting on k-sets, correcting (Cherlin in Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars, 1996---1999, pp. 15---48, Birkhauser 2000, Example 5).

Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. I

This is the first in a series of papers whose results imply the validity of a strengthened version of the Sims conjecture on finite primitive permutation groups from the authors’ article “Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture”, Dokl. Math. 59 (1), 113–115 (1999). In this paper, the case of not almost simple primitive groups and the case of primitive groups with alternating socle are considered.

Invariable generation of permutation groups

Let G be a finite permutation group of degree n, and let d = 2 if G = Sym(3), d = [n/2] otherwise. We prove that there exist d elements g 1, . . . , g d in G with the property that $${G=\langle g_1^{x_1},\ldots,g_d^{x_d}\rangle}$$ for every choice of $${(x_1,\ldots,x_d)\in G^d}$$ .

An atlas of subgroup lattices of finite almost simple groups

We provide algorithms to compute and produce subgroup lattices of finite permutation groups. We discuss the problem of naming groups and we propose an algorithm that automatizes the naming of groups, together with possible ways of refinement. Finally we announce an atlas of subgroup lattices for a large collection of finite almost simple groups made available online.

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.