The projective geometry of a group

Wolfgang Bertram (1)

1 IECN - Institut Élie Cartan de Nancy

We show that the pair given by the power set and by the ''Grassmannian'' (set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize several results from preceding joint work with M. Kinyon (arxiv: math.RA/0903.5441), which concerned abelian groups, to the case of general non-abelian groups. Most notably, pairs of subgroups parametrize torsor and semitorsor structures on the power set. The rôle of associative algebras and -pairs from loc. cit. is now taken by analogs of near-rings.

Mots clés: Grassmannian; projective space; relation; torsor (heap; groud; principal homogeneous space); semitorsor; near-ring; generalized lattice

Domaines: Théorie des groupes [math.GR]

Wolfgang Bertram. The projective geometry of a group. 2012. ⟨hal-00664200⟩