  In this paper we study the parameterized complexity of two well-known
permutation group problems which are NP-complete. 1. Given a permutation group
G=<S>, subgroup of $S_n$, and a parameter $k$, find a permutation $\pi$ in G
such that $|{i\in [n]\mid \pi(i)\ne i}|$ is at least $k$. This generalizes the
well-known NP-complete problem of finding a fixed-point free permutation in G.
(this is the case when $k=n$). We show that this problem with parameter $k$ is
fixed parameter tractable. In the process, we give a simple deterministic
polynomial-time algorithm for finding a fixed point free element in a
transitive permutation group, answering an open question of Cameron. 2. Next we
consider the problem of computing a base for a permutation group G=<S>. A base
for G is a subset B of $[n]$ such that the subgroup of G that fixes B pointwise
is trivial. This problem is known to be NP-complete. We show that it is fixed
parameter tractable for the case of cyclic permutation groups and for
permutation groups of constant orbit size. For more general classes of
permutation groups we do not know whether the problem is in FPT or is
W[1]-hard.
