Generalizations of Zassenhaus lemma and Jordan-Hölder theorem for 2−crossed modules

We present a comprehensive generalization of the Zassenhaus Lemma, the Scherier Refinement Theorem, and the Jordan-Hölder Theorem, extending their applicability to both crossed modules and 2−crossed modules. It is discovered that the previously established normality conditions are insufficient for forming quotient objects of 2−subcrossed modules, as demonstrated through an illustrative example, and these conditions are rigorously revised to allow these generalizations. These new conditions now yield results that are categorically accurate. Additionally, the study has led to the derivation of several supplementary results, including isomorphism theorems for 2−crossed modules.