  Here homogenization theory is used to establish a connection between the
symmetries of a periodic elastic structure associated with the microscopic
properties of an elastic material and the material symmetries of the effective,
macroscopic elasticity tensor. Previous results of this type exist but here
more general symmetries on the microscale are considered. Using an explicit
example, we show that it is possible for a material to be fully anisotropic on
the microscale and yet the symmetry group on the macroscale can contain
elements other than plus or minus the identity. Another example demon- strates
that not all material symmetries of the macroscopic elastic tensor are
generated by symmetries of the periodic elastic structure.
