(untitled) (Communications in Mathematical Research, 2019)
In this paper, first we investigate the invariant rings of the finite groups G ≤ GL(n, F_q) generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F_q in the modular case. Then we are concerned with general groups G_i(ω) and G_i(ω)^t named generalized transvection groups where ω is 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.