A class of groups determined by their $3$-$\text{S}$-rings

Abstract

We study the class of groups $G$ satisfying the condition that, for every ordered pair $x,y \in G$, one of the following is true: (1)~$xy=yx$; (2)~$x$ and $y$ are conjugate; (3)~$x^y=x^{-1}$; (4)~$y^x=y^{-1}$. We describe all such groups completely and give a further condition that characterizes these groups in terms of their $3$-$\text{S}$-rings.