Simple groups, permutation groups, and probability

  title={Simple groups, permutation groups, and probability},
  author={Martin W. Liebeck and Aner Shalev},
  journal={Journal of the American Mathematical Society},
In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability, or counting, is not entirely anticipated by the nature of the problem (see [LiSh2], [GSSh]). In this paper we study a variety of problems in finite simple groups… 

Base size and separation number

The concept of a base for a permutation group was introduced by Sims in the 1960s in connection with computational group theory. It has proved to be of theoretical importance as well. For example,

Bases of primitive permutation groups

A base B for a finite permutation group G acting on a set Ω is a subset of Ω with the property that only the identity of G can fix every element of B. In this dissertation, we investigate some

The spread of finite and infinite groups

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as 32 -generation, that


The most important geometric invariant of a degree-n complex rational function f(X) is its monodromy group, which is a set of permutations of n objects. This monodromy group determines several

On base sizes for algebraic groups

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this

Profinite groups with a rational probabilistic zeta function

In this thesis, we investigate the connection between finitely generated profinite groups G and the associated Dirichlet series PG(s) of which the reciprocal is called the probabilistic zeta function

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

Polynomial-time normalizers

It is proved that, given permutation groups G, H \textless= Sym(Omega) such that G is an element of Gamma(d), the normalizer of H in G can be found in polynomial time.



Simple Groups, Probabilistic Methods, and a Conjecture of Kantor and Lubotzky

Abstract We prove that a randomly chosen involution and a randomly chosen additional element of a finite simple groupGgenerateGwith probability →1 as |G|→∞. This confirms a conjecture of Kantor and

Random Generation of Simple Groups by Two Conjugate Elements

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as

On minimal degrees and base sizes of primitive permutation groups

The minimal degree/~ (G) of a primitive permutat ion group G of degree n on a set ~, that is, the smallest number of points moved by any non-identity element of G, has been the subject of

On the Order of Uniprimitive Permutation Groups

One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for the

Residual properties of free groups

  • S. J. Pride
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1974
In recent yeaxs there has been some research done on the following problem. Given a non-cyclic free group F determine those sets C of groups for which F is residually C . To tackle such a problem it

Upper Bounds for the Number of Conjugacy Classes of a Finite Group

Abstract For a finite group G , let k ( G ) denote the number of conjugacy classes of G . We prove that a simple group of Lie type of untwisted rank l over the field of q elements has at most (6 q )

The probability of generating a classical group

In [KL] it is provcd that the probability of two randomly chosen elements of a finite dassical simple group G actually generating G tends to 1 as lei increases. If gEe, let Pu(g) be the probability

Random Permutations: Some Group-Theoretic Aspects

The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group

The Probability of Generating the Symmetric Group

In [2], Dixon considered the following question: " Suppose two permutations are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? "

Residual properties of free groups II

  • S. J. Pride
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 1972
In this paper it is proved that non-abelian free groups are residually (x, y | xm = 1, yn = 1, xk = yh} if and only if min{(m, k), (n, h)} is greater than 1, and not both of (m, k) and (n, h) are 2