On $p$-index extremal groups

The question on connection between the structure of a finite group $G$ and the properties of the indices of elements of $G$ has been a popular research topic for many years. The $p$-index $\mathrm{\mid }{x}^G{\mathrm{\mid }}_p$ of an element $x$ of a group $G$ is the $p$-part of its index $\mathrm{\mid }{x}^G\mathrm{\mid }=\mathrm{\mid }G:C_G(x)\mathrm{\mid }$. The presented short note describes some new results and open problems in this direction, united by the concept of the $p$-index of a group element.