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 $v\equiv 1\mathrm{mod}k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\varphi$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K⋊⟨\varphi ⟩$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_1^{e_1}\dots p_t^{e_t}\ge 3$ 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 $\phi (k{)}^t$ resolvable $(v,k,1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\phi$ is Euler's totient function.

Final version

References in corpus (1)

• Rotational circulant graphs