Fixed Point Ratios Research Articles

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.

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.

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

AbstractIn this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

Component Fusion Image Encryption Method Based on Hybrid Chaotic Model

The current image encryption method is relatively simple, and there is the problem of poor image encryption effect. Based on the hybrid chaotic model, an image encryption method with component fusion is proposed in this paper. The image is mapped by using Arnold cat mapping method. Chaotic sequence is generated by chaotic model, and the original image is scrambled and substituted to achieve image encryption. Through the fixed point ratio, information entropy, gray mean change value, autocorrelation and similarity, the test of encrypted image is completed. Experimental results show that the proposed image encryption method has good performance, high security intensity, and can effectively encrypt the image.

On the uniform domination number of a finite simple group

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $\gamma_u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s \rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $\gamma_u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $\gamma_u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups $G$ with $\gamma_u(G) = 2$. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.

On base sizes for almost simple primitive groups

Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G0=An and Ω is an orbit of subsets or partitions of {1,…,n}, or if G0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ⩽ 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b(G) ⩽ 7 for any almost simple group G in a non-standard action, proving a conjecture of Cameron. The proof is probabilistic and uses bounds on fixed point ratios.

On base sizes for actions of finite classical groups

AbstractLet G be a finite almost simple classical group and let Ω be a faithful primitive non-standard G-set. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ≤ slant c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ≤ 4, or G = U 6 (2) · 2, G ω = U 4 (3) · 2 2 and b(G) = 5. The proof is probabilistic, using bounds on fixed point ratios.

Fixed point ratios in actions of finite classical groups, IV

This is the final paper in a series of four on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr ( x ) ≲ | x G | − 1 2 for all elements x ∈ G of prime order, or ( G , Ω ) is one of a small number of known exceptions. In this paper we assume G ω is either an almost simple irreducible subgroup in Aschbacher's S collection, or a subgroup in a small additional set N which arises when G has socle Sp 4 ( q ) ′ ( q even) or P Ω 8 + ( q ) . This completes the proof of the main theorem.

Fixed point ratios in actions of finite classical groups, III

This is the third in a series of four papers on fixed point ratios for non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr ( x ) ≲ | x G | − 1 2 for all elements x ∈ G of prime order, or ( G , Ω ) is one of a small number of known exceptions. In this paper we consider the case where G ω is contained in one of the Aschbacher families C 2 or C 3 .

Fixed point ratios in actions of finite classical groups, II

This is the second in a series of four papers on fixed point ratios in non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr ( x ) ≲ | x G | − 1 2 for all elements x ∈ G of prime order, or ( G , Ω ) is one of a small number of known exceptions. In this paper we record a number of preliminary results and prove the main theorem in the case where the stabiliser G ω is contained in a maximal non-subspace subgroup which lies in one of the Aschbacher families C i , where 4 ⩽ i ⩽ 8 .

Fixed point ratios in actions of finite classical groups, I

This is the first in a series of four papers on fixed point ratios in actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either fpr ( x ) ≲ | x G | − 1 2 for all elements x ∈ G of prime order, or ( G , Ω ) is one of a small number of known exceptions. Here fpr ( x ) denotes the proportion of points in Ω which are fixed by x. In this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups. A further application concerning monodromy groups of covers of Riemann surfaces is also outlined.

Critical dynamics of superconductors in the charged regime.

We investigate the finite temperature critical dynamics of three-dimensional superconductors in the charged regime, described by a transverse gauge field coupling to the superconducting order parameter. Assuming relaxational dynamics for both the order parameter and the gauge fields, within a dynamical renormalization group scheme, we find a new dynamic universality class characterized by a finite fixed point ratio between the transport coefficients associated with the order parameter and gauge fields, respectively. We find signatures of this universality class in various measurable physical quantities, and in the existence of a universal amplitude ratio formed by a combination of physical quantities.

Fixed point ratios in actions of finite exceptional groups of Lie type

Let G be a finite exceptional group of Lie type acting transitively on a set O. For x in G, the fixed point ratio of x is the proportion of elements of O which are fixed by x. We obtain new bounds for such fixed point ratios. When a point-stabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism.

FIXED POINT RATIOS IN EXCEPTIONAL GROUPS OF RANK AT MOST TWO

ABSTRACT We obtain upper bounds for the fixed point ratios of the faithful primitive representations of those almost simple finite groups G for which is group of exceptional Lie type of Lie rank 1 or 2. These bounds are shown to be either best possible, or, in the case of very close to best possible.

Composition Factors of Monodromy Groups

The authors prove the 1990 conjecture of Guralnick and Thompson on composition factors of monodromy groups. Using Riemann's existence theorem, the conjecture translates into a problem on primitive permutation groups. This group theoretic problem had been reduced to a question about actions of classical groups on subspaces of their natural modules. The key ingredients in the present proof are the authors' earlier fixed point ratio estimates for such actions and a result of Scott on the generation of linear groups.

Grassmannian Fixed Point Ratios

We provide estimates for the fixed point ratios in the permutation representations of a finite classical group over a field of order q on k-subspaces of its natural n-dimensional module. For sufficiently large n, each element must either have a negligible ratio or act linearly with a large eigenspace. We obtain an asymptotic estimate in the latter case, which in most cases is q−dk where d is the codimension of the large eigenspace. The results here are tailored for our forthcoming proof of a conjecture of Guralnick and Thompson on composition factors of monodromy groups.

Character and Fixed Point Ratios in Finite Classical Groups

Character and Fixed Point Ratios in Finite Classical Groups