Isolated quotient singularities in characteristic $p$

The famous theorem of Chevalley, Shephard and Todd states that the quotient variety $V/G$ of a vector space $V$ by a finite linear group $G$ is non-singular if and only if the group $G$ is generated by pseudoreflections. First this theorem was proven in characteristic $0$, and later generalized for groups $G$ whose order is coprime to the characteristic of the base field. In the modular case (characteristic divides the order of $G$) the part “only if” of the theorem is no longer correct. Kemper and Malle proved a theorem that strengthens Chevalley–Shephard–Todd Theorem for irreducible modular groups generated by pseudoreflections. In our report we shall discuss the results of Kemper and Malle, their connection to the problem of classification of isolated quotient singularities in characteristic p, and also we shall present the results of Stepanov and Shchigolev generalizing Kemper–Malle Theorem for reducible groups in dimension 3. As a consequence, we shall see that the classification of isolated modular quotient singularities of dimension not greater than 3 is essentially the same as the classification of isolated non-modular quotient singularities.