Abstract. We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F .The notion of the essential dimension ed(G) of a finite group G over a field F was introduced in [5]. The integer ed(G) is equal to the smallest number of algebraically independent parameters required to define a Galois G-algebras over any field extension of F . If V is a faithful linear representation of G over Prop. 4.15]). The essential dimension of G can be smaller than dim(V ) for every faithful representation V of G over F . For example, we have ed(Z/3Z) = 1 over Q or any field F of characteristic 3 (cf. [2, Cor. 7.5]) and ed(S 3 ) = 1 over C (cf. [5, Th. 6.5]).In this paper we prove that if G is a p-group and F is a field of characteristic different from p containing p-th roots of unity, then ed(G) coincides with the least dimension of a faithful representation of G over F (cf. Theorem 4.1).We also compute the essential p-dimension of a p-group G introduced in [15]. We show that ed p (G) = ed(G) over a field F containing p-th roots of unity.In the paper the word "scheme" means a separated scheme of finite type over a field and "variety" an integral scheme.
Acknowledgment:We are grateful to Zinovy Reichstein for useful conversations and comments.