On the Essential and Generic Dimension of p-Groups in Characteristic p

Buhler and Reichstein introduce the notion of essential dimension for a group G over a field k to be the degree of the smallest transcendental extension of k on which G has a faithful action. The related notion of generic dimension is introduced by DeMeyer to be the least number of parameters needed to construct a G-generic polynomial. A simple exercise shows that a generic polynomial provides an explicit extension on which G acts faithfully, which has transcendence degree equal to the generic dimension. Thus generic dimension gives an upper bound on essential dimensions, and the main conjecture of this theory is that the two coincide (when the former is defined).

In characteristic p, a classical result of the theory of Witt Vectors gives a generic C_Pⁿextension in n variables. We seek to extend a result of Buhler and Reichstein to the case of p-groups in characteristic p.  Given a non-split extension G of a group H by the cyclic group of order p the essential dimension of G is strictly greater than H. In cases where this is true, it is a direct corollary as to what the essential dimension of G is (one more than the exponent of the Frattini subgroup). Moreover in the cyclic case, we have, via Witt, the equality to generic dimension, that is n. We will report on our advances to this end.