Symmetry assumptions on the geometrical framework have provided successful
mechanisms to develop physically meaningful solutions to many problems. In
tele-parallel gravity, invariance of the frame and spin-connection under a
group of motions defines an affine symmetry group. Here, we assume there exists
a three-dimensional group of affine symmetries acting simply transitively on a
spatial hypersurface and that this group of symmetry actions defines our affine
frame symmetry group. We determine the general form of the co-frame and spin
connection for each spatially homogeneous Bianchi type. We then construct the
corresponding field equations for f(T) tele-parallel gravity. We show that if
the symmetry group is of Bianchi type A (I, II, VI0β, VII0β, VIII or
IX) then there exists a co-frame/spin connection pair that is consistent with
the antisymmetric part of the field equations of f(T) tele-parallel gravity.
For those geometries having a Bianchi type B symmetry group (IV, V, VIhβ,
VIIhβ), we find that in general these geometries are inconsistent with the
antisymmetric part of the f(T) tele-parallel gravity field equations unless
the theory reduces to an analog of General Relativity with a cosmological
constant.Comment: 28 page