In the spirit of Beckett:

"Bobbies on bicycles two by two…" — Roger Miller, 1965

The Literary Field

A mathematics weblog in Australia today—

Clearly, the full symmetric group contains elements
with no regular cycles, but what about other groups?  
Siemons and Zalesskii showed that for any group G 
between PSL(n,q) and PGL(n,q) other than for
(n,q)=(2,2) or (2,3), then in any action of G, every
element of G  has a  regular cycle, except G=PSL(4,2)
acting on  8 points.  The exceptions are due to
isomorphisms with the symmetric or alternating groups.