Permutation groups and transformation semigroups: results and problems

Primitive permutation groups with a solvable 2-transitive subconstituent

We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the 2-homogeneous and the primitive groups. Lastly, we define k-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a k-reachable permutation group of degree n with \(6 \le k \le n - 6\) is either the alternating or the symmetric group.

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω∖{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω. 1991 Mathematics Subject Classification 20B07, 20B05.

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.

Primitive permutation IBIS groups

A permutation group G ≤ Sym(X) on a finite set X is sharp if |G|=∏ l∈L(G)(|X| − l), where L(G) = {|fix(g)| | 1 ≠ g ∈ G}. We show that no finite primitive permutation groups of twisted wreath type are sharp.

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega , \mu )$ admits an invariant $\sigma $-finite measure equivalent to $\mu $. Second, we prove the following de Finetti-type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^{M}$, where $Z$ is a Polish space, is a product measure.

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

On the orders of primitive groups

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

Let \(S\) be a transformation semigroup of degree \(n\). To each element \(s\in S\) we associate a permutation group \(G_R(s)\) acting on the image of \(s\), and we find a natural generating set for this group. It turns out that the \(\mathcal{R}\)-class of \(s\) is a disjoint union of certain sets, each having size equal to the size of \(G_R(s)\). As a consequence, we show that two \(\mathcal{R}\)-classes containing elements with equal images have the same size, even if they do not belong to the same \(\mathcal{D}\)-class. By a certain duality process we associate to \(s\) another permutation group \(G_L(s)\) on the image of \(s\), and prove analogous results for the \(\mathcal{L}\)-class of \(S\). Finally we prove that the Schutzenberger group of the \(\mathcal{H}\)-class of \(s\) is isomorphic to the intersection of \(G_R(s)\) and \(G_L(s)\). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$, or $G$ belongs to explicit families of examples.

Bases for Primitive Permutation Groups and a Conjecture of Babai

Extremely primitive classical groups

Transitive primitive permutation groups acting on trees