English

Resolvable Mendelsohn designs and finite Frobenius groups

Combinatorics 2018-09-27 v3

Abstract

We prove the existence and give constructions of a (p(k)1)(p(k)-1)-fold perfect resolvable (v,k,1)(v, k, 1)-Mendelsohn design for any integers v>k2v > k \ge 2 with v1modkv \equiv 1 \mod k such that there exists a finite Frobenius group whose kernel KK has order vv and whose complement contains an element ϕ\phi of order kk, where p(k)p(k) is the least prime factor of kk. Such a design admits KϕK \rtimes \langle \phi \rangle as a group of automorphisms and is perfect when kk is a prime. As an application we prove that for any integer v=p1e1ptet3v = p_{1}^{e_1} \ldots p_{t}^{e_t} \ge 3 in prime factorization, and any prime kk dividing piei1p_{i}^{e_i} - 1 for 1it1 \le i \le t, there exists a resolvable perfect (v,k,1)(v, k, 1)-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if kk is even and divides pi1p_{i} - 1 for 1it1 \le i \le t, then there are at least φ(k)t\varphi(k)^t resolvable (v,k,1)(v, k, 1)-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where φ\varphi is Euler's totient function.

Keywords

Cite

@article{arxiv.1307.7455,
  title  = {Resolvable Mendelsohn designs and finite Frobenius groups},
  author = {D. F. Hsu and Sanming Zhou},
  journal= {arXiv preprint arXiv:1307.7455},
  year   = {2018}
}

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