Let t be an involution in GL(n, q) whose fixed point space E+ has dimension k between n/3 and 2n/3. For each g is an element of GL(n, q) such that tt(9) has even order, < tt(g)> contains a unique involution z(g) which commutes with t. We prove that, with probability at least c/log n (for some c > 0), the restriction z(g)(vertical bar E+) is an involution on E+ with fixed point space of dimension between k/3 and 2k/3. This result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic. We discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups. (C) 2017 Elsevier Inc. All rights reserved.

In 1973, Charles Sims [89] proved the existence of the Lyons–Sims sporadic simple group Ly by constructing its action as a group of permutations of a set of cardinality 8,835,156 on a computer which could not even store and multiply the two generators of Ly in this smallest degree permutation representation for the group! The existence of this finite simple group, together with many of its properties, had been predicted by Richard Lyons [60], but proof of existence was not established until Sims’ construction. Leading up to this seminal achievement, Sims [88] had developed concepts and computational methods that laid the foundation for his general theory of permutation group computation.

We study s-arc-transitive graphs with s⩾2, and give a characterisation of the actions of vertex-transitive normal subgroups. An interesting consequence of this characterisation states that each non-bipartite 3-arc-transitive graph is a normal cover of a 2-arc-transitive graph admitting a simple group, or a locally primitive graph admitting a simple group with a soluble vertex stabiliser.

Let $g$, $h$ be a random pair of generators of $G=Sym(n)$ or $G=Alt(n)$. We show that, with probability tending to $1$ as $n\to \infty$, (a) the diameter of $G$ with respect to $S = \{g,h,g^{-1},h^{-1}\}$ is at most $O(n^2 (\log n)^c)$, and (b) the mixing time of $G$ with respect to $S$ is at most $O(n^3 (\log n)^c)$. (Both $c$ and the implied constants are absolute.) These bounds are far lower than the strongest worst-case bounds known (in Helfgott--Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap. Our results rest on a combination of the algorithm in (Babai--Beals--Seress, 2004) and the fact that the action of a pair of random permutations is almost certain to act as an expander on $\ell$-tuples, where $\ell$ is an arbitrary constant (Friedman et al., 1998).

An s-geodesic in a graph Γ is a path connecting two vertices at distance s. Being locally transitive on s-geodesics is not a monotone property: if an automorphism group G of a graph Γ is locally transitive on s-geodesics, it does not follow that G is locally transitive on shorter geodesics. In this paper, we characterise all graphs that are locally transitive on 2-geodesics, but not locally transitive on 1-geodesics.

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.

Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G,A))$ of the Cayley graph $\Gamma(G,A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam(G):= $\max_A$ diam$(\Gamma(G,A))$. It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$, but the best previously known upper bound was exponential in $\sqrt{n \log n}$. We give a quasipolynomial upper bound, namely, \[\text{diam}(G) = \exp(O((\log n)^4 \log\log n)) = \exp((\log \log |G|)^{O(1)})\] for G = Sym(n) or G = \Alt(n), where the implied constants are absolute. This addresses a key open case of Babai's conjecture on diameters of simple groups. By standard results, our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree $n$.

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within K n , building up a simple graph. The edges must be chosen according to a set of restrictions \({\mathcal{R}}\) . The winner is the last player to choose an edge that does not violate any of the restrictions in \({\mathcal{R}}\) . For fixed n and \({\mathcal{R}}\) , one of the players has a winning strategy. For various games where \({\mathcal{R}}\) includes bounded degree and triangle avoidance, we determine the winner for all values of n.

Gallai-colorings of complete graphsedge colorings such that no triangle is colored with three distinct colorsoccur in various contexts such as the theory of partially ordered sets (in Gallai's original paper), information theory and the theory of perfect graphs. A basic property of Gallai-colorings with at least three colors is that at least one of the color classes must span a disconnected graph. We are interested here in whether this or a similar property remains true if we consider colorings that do not contain a rainbow copy of a fixed graph F. We show that such graphs F are very close to bipartite graphs, namely, they can be made bipartite by the removal of at most one edge. We also extend Gallai's property for two infinite families and show that it also holds when F is a path with at most six vertices.

The sporadic simple group Monster, denoted by M, acts on the Griess algebra, which is a real vector space of dimension 196,884, equipped with a positive definite scalar product and a bilinear, commutative, and non-associative algebra product. Certain properties of this linear representation of M, together with properties (discovered by Conway and Miyamoto) of idempotents in the Griess algebra that correspond to 2A involutions in M, have been defined by Ivanov as the M-representation of the Monster. This definition enables us to talk about M-representations of arbitrary groups G that are generated by involutions. In general, an M-representation may or may not exist, but if G is isomorphic to a subgroup of the Monster and a representation is isomorphic to the corresponding subalgebra of the Griess algebra then we say that the M-representation is based on an embedding of G in the Monster. In this paper, we describe a generic theoretical procedure to construct M-representations, and a GAP computer program that implements the procedure. It turns out that in many cases the representations are based on embeddings in the Monster, thereby providing a valuable tool of studying subalgebras of the Griess algebra that were unaccessible in the 196,884-dimensional setting.

A non-regular primitive permutation group is said to be extremely primitive if a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In a recent paper, we classified the extremely primitive almost simple classical groups, and in this note we determine the examples with a sporadic or alternating socle. We obtain two infinite families for A(n) (or S-n); they comprise the natural 2-primitive action of n points, plus the action on partitions of {1, ... , n} into subsets of size n/2 (with n/2 odd). There are twenty examples for sporadic groups, including the rank 6 representation of Co-2 on the cosets of McL.

A lambda-design is a family B1,B2,...,Bv of subsets of X={1,2,...,v} such that vertical bar BinBj vertical bar=lambda for all i lambda j and not all Bi are of the same size. Ryser's and Woodall's ?-design conjecture states that each lambda-design can be obtained from a symmetric block design by a certain complementation procedure. Our main result is that the conjecture is true when lambda < 63. (c)(c) 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 408431, 2012

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A near-polygonal graph is a graph Γ which has a set \({\mathcal{C}}\) of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in \({\mathcal{C}}\) . If m is the girth of Γ then the graph is called polygonal. We provide a construction for an infinite family of 2-arc transitive near-polygonal graphs of valency 10 and provide a method for determining the length m of special cycles in graphs in this family. This provides us with some new instances of polygonal graphs as well.

A primitive permutation group is said to be extremely primitive if it is not regular and a point stabilizer acts primitively on each of its orbits. By a theorem of Mann and the second and third authors, every finite extremely primitive group is either almost simple or of affine type. In this paper, we determine the examples in the case of almost simple classical groups. They comprise the 2-transitive actions of PSL2(q) and its extensions of degree q+1, and of Sp2m(2) of degrees 22m−1±2m−1, together with the 3/2-transitive actions of PSL2(q) on cosets of Dq+1, with q+1 a Fermat prime. In addition to these three families, there are four individual examples.

Suppose Γ is a group acting on a set X , written as ( Γ,X ). An r -labeling f: X →{1,2, ..., r } of X is called distinguishing for ( Γ,X ) if for all σ∈Γ,σ ≠1, there exists an element x ∈ X such that f ( x )≠ f ( x σ ). The distinguishing number d ( Γ,X ) of ( Γ,X ) is the minimum r for which there is a distinguishing r -labeling for ( Γ,X ). If Γ is the automorphism group of a graph G , then d ( Γ,V ( G )) is denoted by d ( G ), and is called the distinguishing number of the graph G. The distinguishing set of Γ -actions is defined to be D *( Γ )={ d ( Γ,X ): Γ acts on X }, and the distinguishing set of Γ -graphs is defined to be D ( Γ )={ d ( G ): Aut( G )≅ Γ }. This paper determines the distinguishing set of Γ -actions and the distinguishing set of Γ -graphs for almost simple groups Γ.

We describe an algorithm to compute tensor decompositions of central products of groups. The novelty over previous algorithms is that in the case of matrix groups that are both tensor decomposable and imprimitive, the new algorithm more often outputs the more desirable tensor decomposition.

Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.

We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on $n$ vertices. The two players take turns choosing edges within $K_{n}$, building up a simple graph. The edges must be chosen according to a set of restrictions $\mathcal{R}$. The winner is the last player to choose an edge that does not violate any of the restrictions in $\mathcal{R}$. For fixed $n$ and $\mathcal{R}$, one of the players has a winning strategy. For a pair of games where $\mathcal{R}$ includes bounded degree, connectedness, and triangle-avoidance, we determine the winner for all values of $n$.

The Monster group M , which is the largest among the 26 sporadic simple groups is the automorphism group of the 196,884-dimensional Conway–Griess–Norton algebra (simply called the Monster algebra). There is a remarkable correspondence between the so-called 2 A -involutions in M and certain idempotents in the Monster algebra (we refer to these idempotents as Majorana axes). The isomorphism types of the subalgebras in the Monster algebra generated by pairs of Majorana axes were calculated by S. Norton a while ago (there are precisely nine isomorphism types). More recently these nine algebras were characterized by S. Sakuma in the context of Vertex Operator Algebras, relying on earlier work by M. Miyamoto. The properties of Monster algebras used in the proof of Sakuma’s theorem are rather elementary and they have been axiomatized under the name of Majorana representations. In this terminology Sakuma’s theorem amounts to classification of the Majorana representations of the dihedral groups together with a remark that all the representations are based on embeddings into the Monster. In the present paper it is shown that the alternating group A 5 of degree 5 possesses precisely two Majorana representations, both based on embeddings into the Monster. The dimensions of the representations are 20 and 26; the scalar squares of their identities are 10 and 72/7, respectively (in the Vertex Operator Algebra context these numbers are doubled central charges).