Abstract
The paper continues the study of the orthogonal group of a free symmetric inner product space V over a full ring R which was initiated in [4]. If 2 and 3 are units in R and V has dimension 3 or ⩾5 and hyperbolic rank ⩾1, then the normal subgroups of the orthogonal group of V lie between congruence subgroups. This paper also examines the theory of the special orthogonal group SO( V) of a symmetric inner product space V over a full ring R with 2 a unit. If V has hyperbolic rank ⩾1 and dimension ⩾3, we determine generators, commutator subgroups and the special congruence subgroups. Further, SO( V) is related to the Clifford algebra of V and the normal subgroups of SO( V) are described.
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