Date: Thu, 19 Mar 1998 14:59:32 +0200 (IST)
Reply-To: Assaf Libman 
Subject: Endomorphisms of orthogonal groups


I am interested in the following problem and would appreciate any hint or 
help.

Let f an n-ary non-degenerate, positive definite quadratic form on the
field of rational numbers Q.
Let G=P\Omega_n(Q,f), namely G is the commutator subgroup of the
orthogonal group O_n(Q,f) modulo its center.
It is well known that for n>8, this group is simple.
Is it true that any embedding G-->G is an automorphism ? (namely it is
surjective).
More generally, can one characterize the groups for which the
endomorphisms ring consists only of their automorphisms and the trivial
homomorphism ?


Assaf Libman