The structure of symplectic groups over arbitrary commutative rings

LetR be an arbitrary commutative ring, andn be an integer ≥3. It is proved for any idealJ ofR thatESp _2n(R,J)=[ESp _2n(R),ESp _2n(J)]=[ESp _2n(R),ESp _2n(R,J)] =[ESp _2n(R),GSp _2n(R,J)]=[Sp _2n(R),ESp _2n(R,J)]. Furthermore, the problem of normal subgroups ofSp _2n(R) has an affirmative solution if and only ifaR=a ²R+2aR for eacha inR. This covers the relevant results of [4], [8], [10], [12] and [13]. Project Supported by the Science Fund of the Chinese Academy of Sciences