Steinberg groups and simplicity of elementary groups defined by Jordan pairs

In this paper we study the projective elementary group of a nondegenerate Jordan pair with descending chain condition on principal inner ideals. Let V be a simple such Jordan pair. Our main results are:

With exactly 3 exceptions over the Galois fields F₂ and F₃ , the projective elementary group PE(V) is simple,
with 3 classes of exceptions, again over F₂ and F₃, the central quotient of the elementary automorphism group EA(V) is simple or the product of two simple groups.

The proofs are unified and do not use classification. They rely on Iwasawa's simplicity criterion, the most difficult part (and the cause of the exceptions) being perfectness of the groups. For technical reasons, this is handled in a Steinberg-type group St(V) which for unit-regular Jordan pairs is a central extension of PE(V).

The simplicity results are reminiscent of similar ones for algebraic groups and elementary matrix groups over division rings. Therefore, we compute PE(V) in the most important cases and show that the classical simplicity theorems for elementary orthogonal groups, hyperbolic unitary groups over division rings, and algebraic groups admitting parabolic subgroups with commutative unipotent radical are all special cases of our result. On the other hand, elementary unitary but not hyperbolic groups and algebraic groups of type G₂, F₄ and E₈ are not covered by our results.

This paper has appeared in J. Algebra 186 (1996), 207-234. Reprints are available from

Ottmar Loos < ottmar.loos@uibk.ac.at >