Related papers: Primitive permutation groups of degree 3p

We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of…

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the…

We introduce the notion of primitive elements in arbitrary truncated $p$-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the "points of exact order $N$,"…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

In 2000, L. H\'{e}thelyi and B. K\"{u}lshammer proved that if $p$ is a prime number dividing the order of a finite solvable group $G$, then $G$ has at least $2\sqrt{p-1}$ conjugacy classes. In this paper we show that if $p$ is large, the…

We use methods of combinatorial number theory to prove that, for each $n>1$ and any prime $p$, some homotopy group $\pi_i(SU(n))$ contains an element of order $p^{n-1+ord_p([n/p]!)}$, where $ord_p(m)$ denotes the largest integer $\alpha$…

For the subgroups of the Cremona group $\mathrm{Cr}_3(\mathbb C)$ having the form $(\boldsymbol{\mu}_p)^s$, where $p$ is prime, we obtain an upper bound for $s$. Our bound is sharp if $p\ge 17$.

For modular indecomposable representations of a cyclic group $G$ of prime order $p$ we propose a list of polynomial invariants of degree $\leq 3$ that, together with a simple invariant of degree $p$, separate generic orbits and generate the…

Let k be an algebraically closed field of characteristic p>2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under…

Based on recent successes concerning permutation resolutions of representations by Balmer and Gallauer we define a new invariant of finite groups: the p-permutation dimension. We define this analogously to the global dimension of a ring by…

For prime levels $5 \le p \le 19$, sets of $\Gamma_{0}(p)$-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for $\Gamma_{1}(p)$. An explicit formulation of the permutation…

Let $G$ be the simple group ${\rm PSL}(3,2^p)$, where $p$ is a prime number. For any subgroup $H$ of $G$, we compute the M\"obius function of $H$ in the subgroup lattice of $G$. To this aim, we describe the intersections of maximal…

A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less…

Let $p\equiv 1\,(\mathrm{mod}\,9)$ be a prime number and $\zeta_3$ be a primitive cube root of unity. Then $\mathrm{k}=\mathbb{Q}(\sqrt[3]{p},\zeta_3)$ is a pure metacyclic field with group $\mathrm{Gal}(\mathrm{k}/\mathbb{Q})\simeq S_3$.…

We determine the integral homology of PSL4(Z) in degrees at most 5 and determine its p-part in higher degrees for the primes p>=5. Our method applies to other arithmetic groups; as illustrations we include descriptions of the integral…

Using only undergraduate-level methods, we classify all groups of order $p^4$, where $p$ is an odd prime.

Let $p\equiv1\pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.

We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field $\mathbb{F}_{p^n}$ where $p$ is a prime. In time polynomial in $p$ and $n$, the algorithm either outputs an element that…

We consider an experimental design problem for permutations: given a fixed set $X$, and an integer $t$, construct a list $L$ of permutations of $X$ such that every ordered $t$-tuple of distinct elements of $X$ occurs as a consecutive…

We show that if $A$ is a subset of a group of prime order $p$ such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then $A$ is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and $2A$ contains an arithmetic…