@article{9563ce32b03b4e6bb50019522b57f709, title = "On the converse of Gasch{\"u}tz' complement theorem", abstract = "Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (|N|, |G: H|) = 1. If N is abelian, a theorem of Gasch{\"u}tz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z (N) ∩ N ′ ≠ 1. For metabelian groups N, the condition Z (N) ≠ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gasch{\"u}tz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).", author = "Benjamin Sambale", note = "Funding statement: The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/4-1).", year = "2023", month = sep, day = "1", doi = "10.48550/arXiv.2303.00254", language = "English", volume = "26", pages = "931--949", journal = "Journal of group theory", issn = "1433-5883", publisher = "Walter de Gruyter GmbH", number = "5", }