Structure of the orthogonal group On(V) over L-rings Original Research Article

We study the orthogonal group On(V) on a quadratic module V of rank n with an orthogonal basis over a commutative ring R satisfying some condition which holds for the ring of rational integers Z. The group On(V) will be shown to be a finite group. Indeed we factorize On(V) into a semidirect product of a normal subgroup D and a subroup S, where D = diagonal(ε1, ε2, … ,εn)midεi = ±1 and S = a direct product of a finite number of symmetric groups. Also we determine the order of On(V). Further we show that any element in On(V) is a product of three elements of order 2. Moreover, we prove that On(V) is generated by two elements of order 2 and 2n, respectively, also generated by three elements of order 2.