16 research outputs found
Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups
An orthomorphism is a permutation of for which
is also a permutation on . Eberhard,
Manners, Mrazovi\'c, showed that the number of such orthomorphisms is
for odd and zero otherwise.
In this paper we prove two analogs of these results where is
replaced by (a "multiplicative orthomorphism") or with
(an "exponential orthomorphism"). Namely, we show that no
multiplicative orthomorphisms exist for but that exponential
orthomorphisms exist whenever is twice a prime such that is
squarefree. In the latter case we then estimate the number of exponential
orthomorphisms.Comment: 11 pages, 1 figur
Resolvable Mendelsohn designs and finite Frobenius groups
We prove the existence and give constructions of a -fold perfect
resolvable -Mendelsohn design for any integers with such that there exists a finite Frobenius group whose kernel
has order and whose complement contains an element of order ,
where is the least prime factor of . Such a design admits as a group of automorphisms and is perfect when is a
prime. As an application we prove that for any integer in prime factorization, and any prime dividing
for , there exists a resolvable perfect -Mendelsohn design that admits a Frobenius group as a group of
automorphisms. We also prove that, if is even and divides for
, then there are at least resolvable -Mendelsohn designs that admit a Frobenius group as a group of
automorphisms, where is Euler's totient function.Comment: Final versio