On the tensor product of permutation modules

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.

We survey polynomial time algorithms (both deterministic and random) for computations with permutation groups. Particular emphasis is given to algorithms with running time of the form O(n log c |G|), where G is a permutation group of degree n. In the case of small-base groups, i.e., when log |G| is polylogarithmic as a function of n, such algorithms run in nearly linear, O(n logc' n) time. Important classes of groups, including all permutation representations of simple groups except the alternating ones, as well as most primitive groups, belong to this category. For large n, the majority of practical computations is carried out on small-base groups.

Primitive permutation IBIS groups

In this note, we consider transitive permutation groups of degree 2 p , where p is an odd prime, that admit blocks of imprimitivity of size 2 but no blocks of imprimitivity of size p .

Variants of some of the Brauer-Fowler theorems

In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order. In this article we prove that each minimal normal subgroup of an elusive permutation group with square-free degree is abelian. As an almost immediate consequence, it follows that there are no elusive permutation groups of degree 2pq where p, q are distinct odd primes.

Invariant theory for wreath product 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.

This paper surveys some results in the area of maximal subgroups of the finite simple groups and their automorphism groups. The first two sections are concerned with maximal subgroups of the alternating and symmetric groups. We outline the Reduction Theorem and discuss the maximality of primitive groups in the corresponding symmetric groups. In the third section we consider the corresponding questions for the classical groups. The next section contains a brief survey of knowledge of the maximal subgroups of the other almost simple groups. Finally, the last section outlines some recent work on primitive permutation groups of special degrees.

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.

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems.

Generation and random generation: From simple groups to maximal subgroups

Given a finite group $G,$ let $e(G)$ be the expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all of the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa $, where $\kappa $ is an absolute constant that is explicitly described in terms of the Riemann zeta function and is the best possible in this context. Approximately, $\kappa $ equals 2.752394. If $G$ is a permutation group of degree $n,$ then either $G={Sym} (3)$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor +\kappa ^*$ with $\kappa ^* \sim 1.606695.$ These results improve weaker bounds recently obtained by Lucchini.

Let G be a permutation group on a finite set f2 of size n. Then G acts naturally on the set P (f2) of all subsets of f2. In this note we shall show that if G is primitive on f2 and A, $ G then in all except finitely many cases G has a regular orbit on P (O). We were led to consider this question by some recent work of Siemons and Wagner [5] and Inglis [4], where the following situation was considered. Given permutation groups G and H on the same finite set f2, write G ~ H if G and H have the same orbits on P (f2). It is shown in [4] and [5] that if G is primitive, A, ~ G and G ~ H then with a few explicitly known exceptions the same primes divide [GI and [HI. It is an immediate consequence of our result that under the same hypotheses with finitely many exceptions in fact G = H. (Indeed, with the weaker assumption that the longest orbits of G and H on P (f2) have the same length, we can conclude that [ G I = I HI with only finitely many exceptions.) Unlike the arguments in [4] and [5], ours require the classification of finite simple groups. We do however get a good deal of information even without the classification and can obtain our conclusion by elementary methods if G is 2-transitive on f2. The problem of determining pairs G and H of groups with G ~ H without the assumption of primitivity seems more difficult. Note, however, that if G ~ H and G is primitive then so is H: for G is primitive if and only if the only orbits of G on P (O) whose members partition f2 are the trivial ones. Similarly, but more easily, if G ~ H and G is transitive then so is H. We start with a preliminary well-known result. Recall that the minimal degree m of a permutation group G is the least number of points moved by a non-identity element of G, while the base size b is the least number of points of which the stabilizer is the identity.