Essential dimension of finite \(p\)-groups.
Let \(F\) be a field and \(G\) a finite group. The essential dimension \(\text{ed}_F(G)\) of \(G\) over \(F\) is defined to be the smallest integer \(d\geq 0\) such that one requires \(d\) algebraically independent parameters to define a Galois \(G\)-algebra over any finite extension of \(F\). This notion was introduced by \textit{J. P. Buhler} and \textit{Z. Reichstein} in their paper ``On essential dimension of a finite group'', [Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003)]. The computation of \(\text{ed}_F(G)\) is a hard problem for even specific finite groups. \textit{M. Florence}, in a recent paper [Invent. Math. 171, No. 1, 175--189 (2008; Zbl 1136.14035)], calculated the exact value of \(\text{ed}_F(C)\) for a cyclic \(p\) group \(C\), assuming that \(F\) contains a primitive \(p\)th root of unity and subsequently \textit{P. Brosnan, Z. Reichstein} and \textit{A. Vistoli} computed \(\text{ed}_F(G)\) for a class of nonabelian \(p\)-groups [''Essential dimension and algebraic stacks'', preprint, \url{arXiv:math/0701903}]. In the paper under review the authors prove a generalization of these results. Theorem 4.1: Let \(G\) be a finite \(p\)-group and \(F\) be a field containing a primitive \(p\)th root of unity. Then \(\text{ed}_F(G)\) is the smallest integer \(n\) such that \(G\) has a faithful representation \(G\rightarrow \text{GL}_n(F)\). The authors prove as a corollary (Theorem 5.2) that \[ \text{ed}_F(G_1\times G_2)=\text{ed}_F(G_1)+\text{ed}_F(G_2) \] for any two \(p\)-groups \(G_1\) and \(G_2\). Some results of independent interest on central simple algebras and Brauer-Severi varieties are also proven in the paper.
scientific article; zbMATH DE number 5279042
scientific article