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

A group Gamma is said to be CCII if the quotient Gamma/Z(Gamma) is 2-Engel or, equivalently, commutator-inversion invariant, where Z(Gamma) is the center of Gamma. 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.