対称空間の代数的モデルとしての準群(III)

In this paper, we observe the fact that symmetric loops treated in the previous papers [1] and [2] are in a special class of homogeneous loops of [3]. It is shown that the homogeneous structures on symmetric loops are in one-to-one correspondence to quasigroups of reflection. Following N. Nobusawa [5], we consider abelian quasigroups of reflection and show that they correspond to homogeneous structures of a certain class of abelian groups. We give also an example of finite symmetric loop of 27 elements due to [5] . In conclusion of this series of notes we give some geometric observations on symmetric loops as affine symmetric spaces, when the natural differentiable structures are assumed on them. For this purpose we consider symmetric Lie loops of [3]. Then, by applying the results of [3] and [4], it will be seen that Lie triple systems can be regarded as the tangent algebras of symmetuc Lie loops.