2-powered twisted subgroup

Definition

A subset $S$ of a group $G$ is termed a 2-powered twisted subgroup if it satisfies both the following conditions:

1. For every $x \in S$, there exists a unique element $y \in S$ such that $y^2 = x$.
2. $S$ is a twisted subgroup of $G$, i.e., it contains the identity element, is closed under taking inverses, and for every $x,y \in S$, we have $xyx \in S$.

Particular cases

• We can restrict the twisted multiplication on a 2-powered group to any 2-powered twisted subgroup.
• 2-powered twisted subgroups in Baer Lie groups are in correspondence with 2-powered subgroups of the additive group of the Lie ring under the Baer correspondence.