Abstract:
Let Gn;m be a nilpotent subgroup of the odd-dimensional orthogonal group of size 2n+2m+1 over a polynomial ring R; where R is an euclidean domain with Char(R) 6= 2: In this thesis, we inspect the structure of Gn;m: We study a decomposition of Gn;m; namely, the semi direct product. Moreover, we find the upper central series of G1;m and G2;m; followed by the upper central series of Gn;m; for any n: We learn that Gn;m has a nilpotency class of 2n - 1; which is a positive odd-integer and independent of m. Then we find that the upper central series and the lower central series of Gn;m coincide, when n = 1. Further, when n = 2; the upper and the lower central series of Gn;m coincide under certain conditions, whereas when n > 2; the upper central series and the lower central series do not coincide. In this thesis, we present abelian subgroups of Gn;m, for each n: Furthermore, we introduce a formula which finds the inverse of a uni-upper triangular matrix over an integral domain.
