Related papers: Difference Methods and Ferrero Pairs

Let $(N,\Phi)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\mathcal{D}(a;b)$, as \[\mathcal{D}(a;b)=\{x\in \Phi(r)+c\mid r\neq 0,\ b\in \Phi(r)+c,\ |(\Phi(r)+c)\cap (\Phi(a)+b)|=1\}.\] We prove that in…

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…

We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $\gamma_2(G) < \mathrm{Z}(G) <…

Difference families are traditionally built using groups as their basis. This paper looks at what sort of generalised difference family constructions could be made, using the standard basis of translation and difference. The main result is…

Recently, a construction of group divisible designs (GDDs) derived from the decoding of quadratic residue (QR) codes was given. In this paper, we extend the idea to obtain a new family of GDDs, which is also involved with a well-known…

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized…

One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and…

Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…

A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over…

Building upon the construction of a Cayley calibration adapted to a complex structure, we introduce a calibration $\Phi$ in $\bigwedge^8 \mathbf R^{16}$ with $|\Phi^2| = 294$. This enables us to show that the product of two orthogonally…

Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y…

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form $n$ over a field $k$ of characteristic not two, and a category arising from an action…

We study the structure of a family of algebras which encodes a generalization of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to…

If $F$ is a polynomial automorphism over a finite field $\F_q$ in dimension $n$, then it induces a bijection $\pi_{q^r}(F)$ of $(\F_{q^r})^n$ for every $r\in \N^*$. We say that $F$ can be `mimicked' by elements of a certain group of…

Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct…

For any abstract subfactor planar algebra $P$, there exists a finite index extremal subfactor $M_0 \subset M_1$ with $P$ as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and…

We provide structural criteria for some finite factorised groups $G = AB$ when the conjugacy class sizes in $G$ of certain $\pi$-elements in $A\cup B$ are either $\pi$-numbers or $\pi'$-numbers, for a set of primes $\pi$. In particular, we…

We introduce the relative units-Picard complex of an arbitrary morphism of schemes and apply it to the problem of describing the (cohomological) Brauer group of a (fiber) product of schemes in terms of the Brauer groups of the factors.…

Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\Phi(N)=I(R,N)RG$ where $I(R,N)$ is…