  We discuss three applications of efficient quantum algorithms to determining
properties of permutations and group automorphisms. The first uses the
Bernstein-Vazirani algorithm to determine an unknown homomorphism from
$Z_{p-1}^{m}$ to $Aut(Z_{p})$ where $p$ is prime. The remaining two make use of
modifications of the Grover search algorithm. The first finds the fixed point
of a permutation or an automorphism (assuming it has only one besides the
identity). It can be generalized to find cycles of a specified size for
permutations or orbits of a specified size for automorphisms. The second finds
which of a set of permutations or automorphisms maps one particular element of
a set or group onto another. This has relevance to the conjugacy problem for
groups. We show how two of these algorithms can be implemented via programmable
quantum processors. This approach opens new perspectives in quantum information
processing, wherein both the data and the programs are represented by states of
quantum registers. In particular, quantum programs that specify control over
data can be treated using methods of quantum information theory.
