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

1. 16p, where $p\ne 3,7$ is a prime,

2. $4p^3,$ where $p\ne 3$ is a prime,

3. 12pq, where $5\le p<q$ are primes, $3p<q$ and either $p\ne 5$ or $q\ge 30,$

4. 18pq, where $5\le p<q$ are primes and $2p<q,$

5. $p^{2}qrs,$ where $2<p<q<r<s$ are primes, and $(s-1,r)=1$ or qr < s, and

6. 4pqrs, where $3<p<q<r<s$ are primes, pqr < s, $5·7·13\nmid \mathit{pqrs}$ and $5·31·41\nmid \mathit{pqrs}.$

As a corollary, a 2-closed transitive permutation group of degree $d\le 100$ and different from 72 and 96 contains semiregular elements.

Keywords:

• 2-closed permutation group
• elusive
• polycirculant conjecture

2020 Mathematics Subject Classification:

• 20B05
• 20B10

Additional information

Funding

The work of Dragan Marušič is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0062, J1-9108, J1-1694, J1-1695, N1-0140 and J1-2451).