  We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET
MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP
generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION,
and COSET INTERSECTION. No polynomial-time classical algorithms are known for
these problems. We show that for solvable groups, there exist efficient quantum
algorithms for GROUP INTERSECTION if one of the underlying solvable groups has
a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one
of the underlying solvable groups is smoothly solvable. We also study the
decision versions of STABILIZER and ORBIT COSET, which generalizes GROUP
INTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they
reduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE
COSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof
systems.
