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

Let R be an arbitrary commutative ring, and n an integer≥3. It is proved for any ideal J of R that EO_(2n)(R, J)=[EO_(2n)(R), EO_(2n)(J)]=[EO_(2n)(R), EO_(2n)(R, J)] =[EO_(2n)(R), O_(2n)(R,J)]=[O_(2n)(R), EO_(2n)(R,J)]. In particular, EO_(2n)(R, J) is a normal subgroupof O_(2n)(R). Furthermore, the problem of normal subgroups of O_(2n)(R) has an affirmative solution if and only if aR∩ Ann(2)=α~2 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 O_(2n)(R) has an affirmative solution