On strongly real elements of finite groups

Let $G$ be a group and $x\in G$. If $x$ is inverted by an involution $i$ of $G$, i.e., $x^i=x^{-1}$, then the element $x$ is called strongly real. A group consisting of only strongly real elements is called strongly real. In this paper, we study the disposition of strongly real elements in a finite group and the existence of elements that are not strongly real in connection with problems 14.69 and 14.82 from “The Kourovka Notebook.” For the proofs of the theorems, algorithms are created and implemented in the computer algebra system \textsf{GAP4r3}. They can also be applied for some other finite groups.