The Transfer Ideal Under The Action Of A Nonmetacyclic Group In The Modular Case

The construction of the transfer ideal in the modular case is one of main topics of the invariant theory.In this paper,we study the transfer ideal and some properties for the invariant theory for the ring of polynominal Fq[V₄]under the action of a non-metacyclic group P,which is the nonabelian p-group?p?2?of smallest order,in the modular case.Let Fq be a finite field of characteristic p?p? 2?and V₄ a four-dimensional representation space of the nonmetacyclic group P over Fq.We prove that the ring of invariants Fq[V₄]^P is a polynomial algebra by finding a Dade basis for V₄^?,the dual space of the vector space and that the ring of coinvariants Fq[V₄]^P is not the regular representation.By computing the hyperplanes in the vector space V₄ and the orders of their isotropy groups,we obtain the coefficient of 1/?1-t?^n-1in the Laurent expansion of the Pioncaré series of Fq[V₄]^P,and show that the coefficient of 1/?1-t?^n-1 in the Laurent expansion of the Pioncaré series of the ring of invariants has nothing to do with the number of pseudoreflections in the modular case.We prove that the height of the transfer ideal is 1 using the fixed-point sets of the elements of order p in P and that the transfer ideal is a principal ideal.