
\medskip\noindent {\it Determine for which pairs $(v,k)$ there exists a $k$cycle system of order $v$ having an automorphism group of a given type (for instance cyclic) with a given action (for instance regular) on vertices}.
\medskip This problem is often hard but one can consider the following weaker form.
\medskip\noindent {\it Determine for which pairs $(v,k)$ there exists a $k$cycle system of order $v$ having \underline{at least} one automorphism group (no matter of which type!) with a given action on the vertices}.
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I will survey some results on the above problems.
This is joint work with Andrea Burgess and Patrick Niesink.