THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS

Let R be an arbitrary commutative ring, and n an integer≥3. It is proved for any ideal J of R thatEO2n（R, J）=[EO2n（R）, EO2n（J）]=[EO2n（R）, EO2n（R, J）]=[EO2n（R）, O2n（R,J）]=[O2n（R）, EO2n（R,J）].In particular, EO2n（R, J） is a normal subgroupof O2n（R）. Furthermore, the problem of normal subgroups of O2n（R） has an affirmative solution if and only if aR∩ Ann（2）=α2 Ann（2） for each a in R. In particular, if 2 is not a zero divisor in R, then the problem of normal subgroups of O2n（R） has an affirmative solution