2013/07/31 by D. Frank Hsu, D. F. Hsu, Sanming Zhou · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics #Finite Group Theory Research #Frobenius group #Mathematics #Pure mathematics #graph theory and CDMA systems #math.CO #msc:05B05
paper · pdf · open access · doi:10.1017/s0004972718000333
published in Bulletin of the Australian Mathematical Society 98(1), 1-13 (Cambridge University Press) · Final version
openalex publication_date 2018/05/30 · arxiv created 2018/09/26 · arxiv updated 2018/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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≥ 2 with v≡ 1\hspace0.2em\rm mod\hspace0.2em k such that there exists a finite Frobenius group whose kernel K has order v and whose complement contains an element \unicode[STIX]x1D719 of order k , where p(k) is the least prime factor of k . Such a design admits K\rtimes ⟨ \unicode[STIX]x1D719⟩ as a group of automorphisms and is perfect when k is a prime. As an application we prove that for any integer v=p1^e1⋯ pt^et≥ 3 in prime factorisation and any prime k dividing pi^ei-1 for 1≤ i≤ 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 pi-1 for 1≤ i≤ t , then there are at least \unicode[STIX]x1D711(k)t resolvable (v,k,1) -Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where \unicode[STIX]x1D711 is Euler’s totient function.