  We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether $S$ is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
  We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.
