A Jordan-Hölder theorem for crossed squares

Abstract

In this study, we focus on the decomposition of a crossed square through its composition series by generalizing the Jordan-Hölder uniqueness theorem for groups. For this purpose, we discuss and modify the normality conditions on subcrossed squares and extend some results, such as the isomorphism theorems, the Zassenhaus (butterfly) lemma, and the Schreier refinement theorem, to crossed squares.