We compute the automorphism group of the affine surfaces with the coordinate ring isomorphic to a cluster algebra of rank 2.

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ : we call such lattices τ -invariant. These lattices natura...

For a group G with G-conjugacy class of involutions X, the local fusion graph F(G,X) has X as its vertex set, with distinct vertices x and y joined by an edge if, and only if, the product xy has odd order. Here we show that, with only three possible exceptions, for all pairs (G,X) with G a sporadic simple group or the automorphism group of a sporadic simple group, F(G,X) has diameter 2.

In this paper, we study a relative two-weight Z2Z4-additive codes. It is shown that the Gray image of a two-distance Z2Z4-additive code is a binary two-distance code and that the Gray image of a relative two-weight Z2Z4-additive code, with nontrivial binary part, is a linear binary relative two-weight code. The structure of relative two-weight Z2Z4-additive codes are described. Finally, we disc...

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x 6= y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ∆(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(∆(G)) is abelian if and only if ...

A recent result of W. Kantor followed by a work of W. Feit has rekindled interest in the longstanding conjecture of finite cyclic planes. In this paper we prove that the order of the multiplier group equals the odd part of the order of the automorphism group of a Singer group if and only if the order of the plane is 2, 3, or 8. This yields another proof for Feit's result mentioned above.

We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not give designs with new parameters but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.

We consider the problem of uniqueness of certain simultaneity structures in flat spacetime. Absolute simultaneity is specified to be a nontrivial equivalence relation which is invariant under the automorphism group Aut of spacetime. Aut is taken to be the identity-component of either the inhomogeneous Galilei group or the inhomogeneous Lorentz group. Uniqueness of standard simultaneity in the f...

We construct two flag-transitive designs with the parameters of the title, and prove that these are the only two examples. One is pointprimitive and related to unitary geometry, while the other is pointimprimitive and constructed from a one-dimensional affine space. This classification may be contrasted with a construction by Mathon and Spence in 1996 of more than 3700 designs with these parame...