Primitive Permutation Groups Research Articles

Determination of the Degree of Homogeneity of Primitive Permutation Groups via the Socle of the Groups

The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type.

On Gluck’s conjecture for wreath product type groups

Abstract A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 𝐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 𝐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 𝐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree.

New characterizations of primitive permutation groups with applications to synchronizing automata

For a finite permutation group on n elements we show the following (and variants thereof) equivalences: (1) the permutation group is primitive, (2) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every non-empty subset, an element mapping the whole permutation domain onto this subset, (3) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every two distinct subsets, an element mapping precisely one to a singleton set. We also investigate further properties related to the reachability of subsets. Lastly, we apply our results to automata and show that automata whose transformation monoids contain a primitive permutation group and a mapping that excludes precisely one state from its image are completely reachable and have the property that a minimal automaton for the set of synchronizing words has the maximal possible number of states.

Minimal degree, base size, order: selected topics on primitive permutation groups

In this survey article, we discuss the minimal degree, the base size, and the order of a finite primitive permutation group, along the lines of an article by Martin W. Liebeck.

On the Burness-Giudici conjecture

Let G be a permutation group on a set . A subset of is a base for G if its pointwise stabilizer in G is trivial. By b(G) we denote the size of the smallest base of G. Every permutation group with b(G) = 2 contains some regular suborbits. It is conjectured by Burness-Giudici that every primitive permutation group G with b(G) = 2 has the property that if then , where Γ is the union of all regular suborbits of G relative to α. An affirmative answer of the conjecture has been shown for many sporadic simple groups and some alternating groups, but it is still open for simple groups of Lie-type. The first candidate of an infinite family of simple groups of Lie-type we should work on might be , where . In this manuscript, we show the correctness of the conjecture for all the primitive groups with socle , see Theorem 1.3.

Extremely primitive groups and linear spaces

A finite non-regular primitive permutation group G is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphisms that are extremely primitive on points.

On the commuting probability of p-elements in a finite group

Let $G$ be a finite group, let $p$ be a prime and let ${\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with ${\rm Pr}_p(G) = (p^2+p-1)/p^3$ and we classify all such groups. Our proof is based on bounding the proportion of $p$-elements in $G$ that commute with a fixed $p$-element in $G \setminus \textbf{O}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.

Classifying solvable primitive permutation groups of low rank

Classifying solvable primitive permutation groups of low rank

Saxl graphs of primitive affine groups with sporadic point stabilizers

Let [Formula: see text] be a permutation group on a set [Formula: see text]. A base for [Formula: see text] is a subset of [Formula: see text] whose pointwise stabilizer is trivial, and the base size of [Formula: see text] is the minimal cardinality of a base for [Formula: see text]. If [Formula: see text] has base size [Formula: see text], then the corresponding Saxl graph [Formula: see text] has vertex set [Formula: see text] and two vertices are adjacent if and only if they form a base for [Formula: see text]. A recent conjecture of Burness and Giudici states that if [Formula: see text] is a finite primitive permutation group with base size [Formula: see text], then [Formula: see text] has the property that every two vertices have a common neighbour. We investigate this conjecture in the case where [Formula: see text] is an affine group and a point stabilizer is an almost quasisimple group whose central quotient is either [Formula: see text] or [Formula: see text] for some sporadic simple group [Formula: see text]. We verify the conjecture for all but [Formula: see text] of the groups [Formula: see text].

Bounds for the Diameters of Orbital Graphs of Affine Groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer H ≤GL(V ), where the vector space V has dimension d over the prime field, can be bounded in terms of d and log |V|/log |H| only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

Base sizes of primitive groups of diagonal type

Abstract Let G be a permutation group on a finite set $\Omega $ . The base size of G is the minimal size of a subset of $\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.

On the classification of extremely primitive affine groups

Let G be a finite non-regular primitive permutation group on a set Ω with point stabiliser Gα. Then G is said to be extremely primitive if Gα acts primitively on each of its orbits in Ω \ {α}, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collection of groups. Moreover, if one assumes Wall’s conjecture on the number of maximal subgroups of an almost simple group, then there is an explicit list of candidates, each of which has been eliminated in a recent paper by Burness and Thomas. So, modulo Wall’s conjecture, the classification of extremely primitive groups is complete. In this paper we adopt a different approach, which allows us to complete this classification in full generality, independent of the veracity or otherwise of Wall’s conjecture in the almost simple setting. Our method relies on recent work of Fawcett, Lee and others on the existence of regular orbits of almost simple groups acting on irreducible modules.

Fixed point ratios for finite primitive groups and applications

Let G be a finite primitive permutation group on a set Ω and recall that the fixed point ratio of an element x∈G, denoted fpr(x), is the proportion of points in Ω fixed by x. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing fpr(x) with the order of x. Our main theorem provides a classification of the triples (G,Ω,x) as above with the property that x has prime order r and fpr(x)>1/(r+1). There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree m with minimal degree at most 2m/3. Secondly, our main result plays a key role in recent joint work with Moretó and Navarro on the commuting probability of p-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava.

On the orbital diameter of groups of diagonal type

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify all finite simple groups with covering number and conjugacy width at most 3. We also prove some general bounds on the covering number and conjugacy width of groups of Lie type.

Primitive set-theoretic solutions of the Yang–Baxter equation

To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set [Formula: see text] there is a naturally associated finite solvable permutation group [Formula: see text] acting on [Formula: see text]. We prove that every primitive permutation group of this type is of prime order [Formula: see text]. Moreover, [Formula: see text] is then a so-called permutation solution determined by a cycle of length [Formula: see text]. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.

On the number of conjugacy classes of a primitive permutation group

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.

Jordan permutation groups and limits of 𝐷-relations

Abstract We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

Base sizes for primitive groups with soluble stabilisers

Let $G$ be a finite primitive permutation group on a set $\Omega$ with point stabiliser $H$. Recall that a subset of $\Omega$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b(G,H)$, to be the minimal size of a base for $G$. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that $b(G,H) \leqslant 4$ if $G$ is soluble. In this paper we extend Seress' result by proving that $b(G,H) \leqslant 5$ for all finite primitive groups $G$ with a soluble point stabiliser $H$. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that $4$ random elements in $\Omega$ form a base tends to $1$ as $|G|$ tends to infinity.

Permutation groups with restricted stabilizers

Fix a positive integer d and let Γd be the class of finite groups without sections isomorphic to the alternating group Ad. The groups in Γd were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function f such that the base size of any finite primitive group with point stabilizers in Γd is at most f(d). This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of c points lie in Γd. We also show that if G is any permutation group of degree n whose c-point stabilizers lie in Γd, then |G|⩽((1+oc(1))d/e)n−1. This asymptotically extends and improves a dn−1 upper bound on |G| obtained by Babai, Cameron and Pálfy assuming G∈Γd.

Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set varOmega of size n. A subset {mathcal {B}} of varOmega is a base for G if the pointwise stabilizer G_{({mathcal {B}})} is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of mathrm {Sym}(n) is large base if there exist integers m and r ge 1 such that {{,mathrm{Alt},}}(m)^r unlhd G le {{,mathrm{Sym},}}(m)wr {{,mathrm{Sym},}}(r), where the action of {{,mathrm{Sym},}}(m) is on k-element subsets of {1,dots ,m} and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group mathrm {M}_{24} in its natural action on 24 points, or b(G)le lceil log nrceil +1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.