Semiregular elements in transitive 2-closed permutation groups of certain degrees

Abstract

A nonidentity element of a permutation group is said to be semiregular provided all of its cycles in its cycle decomposition are of the same length. It is known that semiregular elements exist in transitive 2-closed permutation groups of square-free degree and in some special cases when the degree is divisible by a square of a prime. In this paper it is shown that semiregular elements exist in transitive 2-closed permutation groups of the following degrees 16p, where is a prime, where is a prime, 12pq, where are primes, and either or 18pq, where are primes and where are primes, and or qr < s, and 4pqrs, where are primes, pqr < s, and As a corollary, a 2-closed transitive permutation group of degree and different from 72 and 96 contains semiregular elements.