Properties of gyrogroups induced by groups whose central quotients being 2-Engel

: A group Γ is said to be CCII if the quotient Γ /Z (Γ) is 2 -Engel or, equivalently, commutator-inversion invariant, where Z (Γ) is the center of Γ . In this article, we prove algebraic and topological properties of gyrogroups that are induced by CCII groups. Then, using a classification of non-abelian groups of order n with n < 32 , we determine all finite CCII groups of order less than 32 .