Classification Of Simple Groups Research Articles

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.

The classification of flag-transitive Steiner 4-designs

Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades also flag-transitive Steiner tdesigns (i.e. flag-transitive t-(v,k,1) designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple group classification, for Steiner t-designs with parameters t > 2 such characterizations have remained challenging open problems for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating back to around 1965). The object of the present paper is to give a complete classification of all flag-transitive Steiner 4-designs. Our result relies on the classification of the finite doubly transitive permutation groups and is a continuation of the author's work [20, 21] on the classification of all flag-transitive Steiner 3-designs.

LOCAL CHARACTERISTIC $p$ COMPLETIONS OF WEAK $BN$-PAIRS

This paper establishes recognition theorems for the finite Lie-type groups defined in odd characteristic $p$. The hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak $BN$-pair of rank 2. The groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic $p$ (a condition satisfied by the finite Lie-type groups defined in characteristic $p$). The results presented will find application in one of the ongoing revisions of the finite simple group classification.

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.

Möbius regular maps

Möbius regular maps are surface embeddings of graphs with doubled edges such that (i) the automorphism group of the embedding acts regularly on flags and (ii) each doubled edge is a centre of a Möbius band on the surface. In the first part of the paper we give an abstract characterisation of Möbius regular maps with a given automorphism group in terms of two dihedral subgroups intersecting in a special way. As an application we exhibit an interesting correspondence between Möbius regular maps of valence 6 and 3-arc-transitive cubic graphs. The second part of the paper deals with constructions of Möbius regular maps on certain classes of simple groups. The main result here is an exact enumeration of all such maps on PSL ( 2 , q ) groups. A number of other results related to the two main topics are presented.

On limit graphs of finite vertex-primitive graphs

The class of all connected vertex-transitive graphs forms a metric space under a natural combinatorially defined metric. In this paper we study graphs which are limit points in this metric space of the subset consisting of all finite graphs that admit a vertex-primitive group of automorphisms. A description of these limit graphs provides a useful description of the possible local structures of generic finite graphs that admit a vertex-primitive automorphism group. We give an analysis of the possible types of these limit graphs, and suggest directions for future research. Some of the analysis relies on the finite simple group classification.

Products of formations of finite groups

In this paper criteria for a product of formations to be X -local, X a class of simple groups, are obtained. Some classical results on products of saturated formations appear as particular cases.

On the Random Generation of Finite Simple Classical Groups#

Let G be a finite simple classical group with natural module of dimension n, and let r, s be primes, not both 2, with r ≤ s. Denote by P r, s (G) the probability that G is generated by two randomly chosen elements of orders r and s. We prove that there is a quadratic function f such that if n ≥ f(s), then P r, s (G) → 1 as |G| → ∞.

Cohomological invariants and R-triviality of adjoint classical groups

Using a cohomological obstruction, we construct examples of absolutely simple adjoint classical groups of type 2A_n with n equivalent to 3 mod 4, C_n or 1D_n with n equivalent to 0 mod 4, which are not R-trivial hence not stably rational.

On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if H≃G for every finite group H such that ω(H)=ω(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.

Fixed point spaces in actions of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0, and let H be a maximal closed non-subspace subgroup of G. Given such a pair (G, H), we obtain a close to best possible upper bound for the ratio dim(xG ? H) / =dim xG, where x ? G is a semisimple or unipotent element of prime order. We apply this result to the study of fixed point spaces.

On primitive overgroups of quasiprimitive permutation groups

A permutation group is said to be quasiprimitive if all its non-trivial normal subgroups are transitive. We investigate pairs ( G, H) of permutation groups of degree n such that G⩽ H⩽ S n with G quasiprimitive and H primitive. An explicit classification of such pairs is obtained except in the cases where the primitive group H is either almost simple or the blow-up of an almost simple group. The theory in these remaining cases is investigated in separate papers. The results depend on the finite simple group classification.

On Minimal Permutation Representations of Classical Simple Groups

We describe minimal permutation representations, i.e., faithful permutation representations of least degree, of classical finite simple groups as groups of automorphisms of simple Lie algebras.

RANDOM (r, s)-GENERATION OF FINITE CLASSICAL GROUPS

A proof is given that for primes r, s, not both 2, and for finite simple classical groups G of sufficiently large rank, the probability that two randomly chosen elements in G of orders r and s generate G tends to 1 as |G| → ∞.

Irreducibility of tensor squares, symmetric squares and alternating squares

We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups & that act irreducibly on all fundamental representations of &.

On isomorphisms of finite Cayley graphs—a survey

The isomorphism problem for Cayley graphs has been extensively investigated over the past 30 years. Recently, substantial progress has been made on the study of this problem, many long-standing open problems have been solved, and many new research problems have arisen. The results obtained, and methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the finite simple group classification, through to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.

Numbers of classes of classical groups

In the paper, the description of the numbers and groups of classes of the simple classical indefinite groups over a fieldK of algebraic numbers in the canonical realizations is completed. A criterion for a simply connected semisimple algebraic group in some givenK-realization to be a one-class group is proved. A criterion for the existence of a one-class quadratic form among the forms ℚ–equivalent to a definite quadratic form inn>-3 variables over the field ℚ of rational numbers is obtained.

Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2pq, where p, q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2pq are classified. This depends on the finite simple group classification. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2pq is a non-Cayley number, where 2<q<p and q and p are primes, if and only if one of p = 1 (mod 4), or q = 1 (mod 4), or p = 1 (mod q), or p = 4q - 1, or (p, q) = (11, 7) or (19, 7) holds.

On the Isomorphism Problem for Finite Cayley Graphs of Bounded Valency

For a subsetSof a groupGsuch that 1∉SandS=S−1, the associated Cayley graph Cay(G,S) is the graph with vertex setGsuch that {x,y} is an edge if and only ifyx−1∈S. Each σ∈Aut(G) induces an isomorphism from Cay(G,S) to the Cayley graph Cay(G,Sσ). For a positive integerm, the groupGis called anm-CI-group if, for all Cayley subsetsSof size at mostm, whenever Cay(G,S) ≅Cay(G,T) there is an element σ∈Aut(G) such thatSσ=T. It is shown that ifGis anm-CI-group for somem≥4, thenG=U×V, where (|U|,|V|) =1,Uis abelian, andVbelongs to an explicitly determined list of groups. Moreover, Sylow subgroups of such groups satisfy some very restrictive conditions. This classification yields, as corollaries, improvements of results of Babai and Frankl. We note that our classification relies on the finite simple group classification.