The Invariant Rings of the Generalized Transvection Groups in the Modular Case

In this paper, first we investigate the invariant rings of the finite groups G ≤ GL(n, Fq ) generated by i-transvections and i-reflections with given invariant subspaces H over a finite field Fq in the modular case. Then we are concerned with general groups Gi(ω) and Gi(ω)t named generalized transvection groups where ωis a k-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay, Gorenstein, complete intersection, polynomial and Poincare series of these rings.