Generalizations of Zassenhaus lemma and Jordan-Hölder theorem for 2-crossed modules (Q6578095)

A crossed module is defined as a triple \((T,G,\partial)\) where \(G\) is a group acting on the group \(T\) and \(\partial \colon T\rightarrow G\) is a group homomorphism satisfying the axioms (i) \(\partial (g.t)=g\partial(t)g^{-1}\) and \(\partial(s).t=sts^-1\) for \(g\in G\) and \(s,t\in T\). The crossed modules introduced by \textit{J. H. C. Whitehead} [Bull. Am. Math. Soc. 55, 453--496 (1949; Zbl 0040.38801); Ann. Math. (2) 49, 610--640 (1948; Zbl 0041.10102)] are among the structures of the higher-dimensional groups and they are categorically equivalent to Cat-1 groups, simplicial groups, strict 2-groups. The crossed modules generalize both the notions of group and normal subgroup.\N\NIn the paper, the authors give some generalisations of the Zassenhaus Lemma, the Scherier Refinement Theorem, and the Jordan-Hölder Theorem to crossed modules and 2-crossed modules, which play an important role in group theory, both in the theory of crossed modules and in the theory of 2-crossed modules..