On Semiregular Elements of Solvable Groups

Cayley numbers with arbitrarily many distinct prime factors

“…Now, for every finite product n of distinct elements from S, we have gcd(n, φ(n)) = 1 and the only quasiprimitive groups of degree n are A n and S n . Now [6,Corollary 4.10] immediately gives that every vertex-transitive digraph of order n is a Cayley digraph.…”

“…There exists an infinite set of primes S such that, for every finite product n of distinct elements from S with n not prime, the only quasiprimitive groups of degree n are the alternating and the symmetric groups. Theorem 1.2 combined with [6,Theorem 4.7] gives the following interesting corollary. Corollary 1.3.…”

Cayley graphs of more than one abelian group

“…See [4,9] for some generalizations of Pálfy's Theorem (Theorem 1.1). In addition to completely answering the question of which groups are CI-groups with respect to every class of combinatorial objects, Pálfy's Theorem has also been used to classify various classes of vertex-transitive graphs [3,7,10], and using these results, the first author with Pablo Spiga [13] showed that there are Cayley numbers with arbitrarily many prime divisors, settling an old problem of Praeger and McKay. Thus Pálfy's Theorem and its generalizations not only have the obvious applications to the isomorphism problem for Cayley objects, but also to classification problems.…”

Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$

“…In particular, Giudici [9] settled the question for quasiprimitive group actions, leaving as one of the main open cases graphs admitting solvable group actions (see [19]). Furthermore, there have also been a number of papers dealing with semiregularity problem for vertex-transitive graphs satisfying certain valency and order restrictions (see, for instance, [3,4,5,6,7,8,9,10,11,13,20,22,24,25]). For example, it is known that every 2-closed group of square-free degree admits semiregular elements (see [7]).…”