  The framework of quantum symmetry reduction is applied to loop quantum
gravity with respect to transitively acting symmetry groups. This allows to
test loop quantum gravity in a large class of minisuperspaces and to
investigate its features - e.g. the discrete volume spectrum - in certain
cosmological regimes. Contrary to previous studies of quantum cosmology
(minisuperspace quantizations) the symmetry reduction is carried out not at the
classical level but on an auxiliary Hilbert space of the quantum theory before
solving the constraints. Therefore, kinematical properties like volume
quantization survive the symmetry reduction. In this first part the kinematical
framework, i.e. implementation of the quantum symmetry reduction and
quantization of Gauss and diffeomorphism constraints, is presented for Bianchi
class A models as well as locally rotationally symmetric and spatially
isotropic closed and flat models.
