Resolvable Mendelsohn designs and finite Frobenius groups

Abstract

We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v, k, 1)$-Mendelsohn design for any integers $v > k \ge 2$ with $v \equiv 1 \mod k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $ϕ$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle ϕ\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3$ in prime factorization, and any prime $k$ dividing $p_{i}^{e_i} - 1$ for $1 \le i \le t$, there exists a resolvable perfect $(v, k, 1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i} - 1$ for $1 \le i \le t$, then there are at least $φ(k)^t$ resolvable $(v, k, 1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $φ$ is Euler's totient function.