2-powered twisted subgroup

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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