Resolvable Mendelsohn designs and finite Frobenius groups

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 $v \equiv 1 \mod k$suchthatthereexistsafiniteFrobeniusgroupwhosekernel$K$hasorder$v$andwhosecomplementcontainsanelement$\phi$oforder$k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K \rtimes \langle \phi \rangle$asagroupofautomorphismsandisperfectwhen$k$ is a prime. As an application we prove that for any integer $v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3$inprimefactorization,andanyprime$k$dividing$p_{i}^{e_i} - 1$for$1 \le i \le t$,thereexistsaresolvableperfect$(v, k, 1)$-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.