On elementary properties of crossed modules

Abstract

A group models a connected CW-space X where only π_1(X) is allowed to be nontrivial. A crossed module in the sense of Whitehead models a connected CW-space X, where only π_1(X) and π_2(X) are allowed to be nontrivial. We show some elementary assertions for crossed modules that are inspired by concepts from group theory: a Jordan-Hölder Theorem for crossed modules, a Zassenhaus lemma, sometimes called "butterfly lemma", and an orbit lemma for crossed modules. Moreover, we describe the simple crossed modules.