Resolvable Mendelsohn designs and finite Frobenius groups

TL;DR

This paper constructs specific Mendelsohn designs with automorphism groups related to Frobenius groups, providing existence proofs and explicit constructions for various parameters, especially when the design parameters satisfy certain divisibility conditions.

Contribution

It introduces new methods to construct resolvable Mendelsohn designs with automorphisms from Frobenius groups, expanding the known classes of such combinatorial designs.

Findings

Existence of $(p(k)-1)$-fold perfect resolvable Mendelsohn designs under Frobenius group conditions.

Explicit constructions for designs with parameters related to prime factorizations of $v$.

Multiple designs with automorphism groups are shown to exist for various divisibility conditions on $k$.

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 $\phi$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle \phi \rangle$ 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 $\varphi(k)^t$ resolvable $(v, k, 1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\varphi$ is Euler's totient function.