Related papers: Primitive permutation groups of degree 3p

Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime…

Let p(n) be the nth prime and p(p(n)) be the nth prime-indexed prime (PIP). The process of taking prime-indexed subsequences of primes can be iterated, and the number of such iterations is the prime-index order. We report empirical evidence…

In this article, we shall generalize a theorem due to Frobenius in group theory, which asserts that if $p$ is a prime and $p^{r}$ divides the order of a finite group, then the number of subgroups of order $p^{r}$ is $\equiv$ 1(mod $p$).…

We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p.

Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least p^d/4, and apply this classification to determine the finite primitive permutation groups of affine type, and…

Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.

For a prime $p>3$, let $D$ be the discriminant of an imaginary quadratic order with $|D|< \frac{4}{\sqrt{3}}\sqrt{p}$. We research the solutions of the class polynomial $H_D(X)$ mod $p$ in $\mathbb{F}_p$ if $D$ is not a quadratic residue in…

Given an odd prime $p$, we identify composition factors of the reduction modulo $p$ of spin irreducible representations of the covering groups of symmetric groups indexed by partitions with 2 parts and find some decomposition numbers.

Let $G$ be a permutation group acting on a set $V$. A partition $\pi$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $\pi$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a…

Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect…

Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of…

We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the…

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an…

We investigate the occurrence of elements of order $p$ in the upper central series of a finite $p$-group.

For non-negative integers $k\leq n$, we prove a combinatorial identity for the $p$-binomial coefficient $\binom{n}{k}_p$ based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for…

In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new…

We study the existence of (unmixed) Beauville structures in finite $p$-groups, where $p$ is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite $p$-groups satisfying certain conditions which…

A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…

We use only addition and multiplication to construct the primitive roots of $p^{k+1}$ from the primitive roots of $p^{k}$, where $p$ is an odd prime and $k$ is at least 2.

For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes…