Groups with essential dimension one. (English) Zbl 1201.12002

Chu, Huah; Hu, Shou-Jen; Kang, Ming-Chang; Zhang, Jiping

Asian J. Math. 12, No. 2, 177-192 (2008).

The concept of the essential dimension of a finite group \(G\) over a field \(K\), denoted by ed\(_K(G)\) was introduced by J. Buhler and Z. Reichstein in [Compos. Math. 106, No. 2, 159–179 (1997; Zbl 0905.12003)], where it is proved that if char \(K\) = 0 and if \(K\) contains all roots of 1, then ed\(_K(G)=1\) if and only if \(G\) is the cyclic group \(\mathbb{Z}_n\) of order \(n > 1\) or \(G\) is the dihedral group \(D_n\) where \(n\) is odd. The authors of the paper under review strengthen this result by relaxing all conditions on \(K\) and finding all pairs \((K,G)\) for which ed\(_K(G)=1\).

Reviewer: Mowaffaq Hajja (Irbid)

MSC:

12F10	Separable extensions, Galois theory
12F20	Transcendental field extensions
12E05	Polynomials in general fields (irreducibility, etc.)

Keywords:

essential dimension; compression of finite group actions; Galois theory; finite subgroups of \(SL_2(K)\)

Citations:

Zbl 0905.12003