Quadratic rank‐one groups and quadratic Jordan division algebras

A rank‐one group X is a group generated by two distinct nilpotent subgroups A and B such that, for each a ∈ A # , there exists a b ∈ B # satisfying A b = B a and vice versa. It has been shown that the notions of a rank‐one group and of a group with a split BN ‐pair of rank 1 are equivalent. Hence all algebraic groups of relative rank 1 and classical groups of Witt index 1 are rank‐one groups. The rank‐one group X = 〈 A , B 〉 is said to be quadratic if there exists a ℤ X ‐module V satisfying [ V , X , X ] ≠ 0 = [ V , A , A ]. In the main result of this paper we classify the quadratic rank‐one groups, that is, we show that there is a one‐to‐one correspondence of such groups with special quadratic Jordan division algebras.