Finite groups of essential dimension one (Q877694)

The notion of essential dimension of a finite group \(G\) over a field \(K\) was introduced by \textit{J. Buhler} and \textit{Z. Reichstein} [Compos. Math. 106, 159--179 (1997; Zbl 0905.12003)]. In the paper under review it is shown that a finite group \(G\) has essential dimension 1 over an infinite field \(K\) if and only if there exists an embedding of \(G\) into \(\text{GL}_2(K)\) such that the image of \(G\) contains no scalar matrices other than the identity. The proof goes by reducing the theorem to the case where \(G\) is a \(p\)-group and then distinguishing between the cases where \(p= 2\) and \(p\neq 2\).