Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

In this paper, we first prove that for $g\in \{3,4\}$, there are infinitely many 3-geodesic transitive but not 3-arc transitive graphs of girth $g$ with arbitrarily large diameter and valency. Then we classify the family of 3-geodesic transitive but not 3-arc transitive graphs of valency 3 and those of valency 4 and girth 4.

We show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabiliser is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss conjecture.

The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. This removes a primality condition from a classical theorem of Jordan. Some applications to monodromy groups are given, and the contributions of Jordan and Marggraff to this topic are briefly discussed.

Let $\Gamma $ be a $G$-vertex-transitive graph and let $(u,v)$ be an arc of $\Gamma $. It is known that if the local action $G_v^{\Gamma (v)}$ (the permutation group induced by $G_v$ on $\Gamma (v)$) is permutation isomorphic to the dihedral group of degree four, then either $|G_{uv}|$ is ‘small’ with respect to the order of $\Gamma $ or $\Gamma $is one of a family of well-understood graphs. In this paper, we generalise this result to a wider class of local actions.

We describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

We consider the wreath product of two permutation groups G≤Sym Γ and H≤Sym Δ as a permutation group acting on the set Π of functions from Δ to Γ. Such groups play an important role in the O’Nan–Scott theory of permutation groups and they also arise as automorphism groups of graph products and codes. Let X be a subgroup of Sym Γ≀Sym Δ. Our main result is that, in a suitable conjugate of X, the subgroup of SymΓ induced by a stabiliser of a coordinate δ∈Δ only depends on the orbit of δ under the induced action of X on Δ. Hence, if X is transitive on Δ, then X can be embedded into the wreath product of the permutation group induced by the stabiliser Xδ on Γ and the permutation group induced by X on Δ. We use this result to describe the case where X is intransitive on Δ and offer an application to error-correcting codes in Hamming graphs.

We characterise regular bipartite locally primitive graphs of order 2pe, where p is prime. We show that either p=2 (this case is known by previous work), or the graph is a binormal Cayley graph or a normal cover of one of the basic locally primitive graphs; these are described in detail.

For a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 212, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. We also prove that every automorphic loop has the antiautomorphic inverse property, and that a right automorphic loop is automorphic if and only if its conjugations are automorphisms.

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilizers. However, beside a well-known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilizers, each significant for its own reason.

Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

Erdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.

Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 .Such a bound is obtained in Section 2.

The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.

We give a qualitative description of the set 𝒪G(H) of overgroups in G of primitive subgroups H of finite alternating and symmetric groups G, and particularly of the maximal overgroups. We then show that certain weak restrictions on the lattice 𝒪G(H) impose strong restrictions on H and its overgroup lattice.

The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

We establish a link between regular subgroups of the affine group and radical circle algebras on the underlying vector space.

An S3-involution graph for a group G is a graph with vertex set a union of conjugacy classes of involutions of G such that two involutions are adjacent if they generate an S3-subgroup in a particular set of conjugacy classes. We investigate such graphs in general and also for the case where G=PSL(2,q).