Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V]~G, has a finite SAGBI basis. We describe two algorithms for constructing a generating set for k[V]~G. We use these methods to analyse k[2V_3]~(U_3) where U_3 is the p-Sylow subgroup of GL_3(F_p) and 2V_3 is the sum of two copies of the canonical representation. We give a generating set for k[2V_3]~(U_3) for p = 3 and prove that the invariants fail to be Cohen-Macaulay for p > 2. We also give a minimal generating set for k[mV_2]~(Z/p) were V_2 is the two-dimensional indecomposable representation of the cyclic group Z/p.
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