Essential dimension of finite p-groups

Essential Dimension of Finite Groups in Prime Characteristic

Let F be a eld of characteristic p > 0 and G be a smooth nite algebraic group over F . We compute the essential dimension edF (G; p) of G at p. That is, we show that edF (G; p) = { 1, if p divides |G|, and 0, otherwise.

On the essential dimension of cyclic p - groups

Let p be a prime number and r ≥ 1 an integer. We compute the essential dimension of Z/pZ over fields of characteristic not p, containing the p-th roots of unity (theorem 3.1). In particular, we have edQ(Z/8Z) = 4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished).

Automorphisms of Pro-p groups of finite virtual cohomological dimension

Let G be a pro-p group of finite cohomological dimension and type FP∞ and T is a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP∞ (in particular it is finitely presented). Moreover we prove that a pro-p group G of type FP∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

Finite Groups with p-Sylow Coverings

Essential dimensions of finite groups

Let K be an arbitrary field and G be a finite group. We will study the essential dimension of G over K, which is denoted by edK(G). A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159–179, Theorem 5.3) is obtained. As a corollary, it can be proved that edK(Sn) ≥ ⌊ 2 ⌋ and edK(An) ≥ 2⌊ n 4 ⌋ for any field K with charK 6= 2, while edK(Sn) ≥ ...