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Abstract. Let G be a semisimple affine algebraic group of inner type over a field F . We write X G for the class of all finite direct products of projective G-homogeneous Fvarieties. We determine the structure of the Chow motives with coefficients in a finite field of the varieties in X G . More precisely, it is known that the motive of any variety in X G decomposes (in a unique way) into a sum of indecomposable motives, and we describe the indecomposable summands which appear in the decompositions.In the case where G is the group PGL A of automorphisms of a given central simple Falgebra A, for any variety in the class X G (which includes the generalized Severi-Brauer varieties of the algebra A) we determine its canonical dimension at any prime p. In particular, we find out which varieties in X G are p-incompressible. If A is a division algebra of degree p n for some n ≥ 0, then the list of p-incompressible varieties includes the generalized Severi-Brauer variety X(p m ; A) of ideals of reduced dimension p m for m = 0, 1, . . . , n.

For a given sequence of integers (n i ) 1 i=1 we consider all the central simple algebras A (over all elds) satisfying the condition ind A i = n i and nd among them an algebra having the biggest torsion in the second Chow group CH 2 of the corresponding Severi-Brauer variety (\biggest" means that it can be mapped epimorphically onto each other).We describe this biggest torsion in a way in general and more explicitly in some important special situations. As an application we prove indecomposability of certain algebras.

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.