TY - JOUR
T1 - On the converse of Gaschütz' complement theorem
AU - Sambale, Benjamin
N1 - Funding statement: The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/4-1).
PY - 2023/9/1
Y1 - 2023/9/1
N2 - 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 Gaschütz 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 Gaschütz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).
AB - 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 Gaschütz 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 Gaschütz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).
UR - http://www.scopus.com/inward/record.url?scp=85163134466&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2303.00254
DO - 10.48550/arXiv.2303.00254
M3 - Article
AN - SCOPUS:85163134466
VL - 26
SP - 931
EP - 949
JO - Journal of group theory
JF - Journal of group theory
SN - 1433-5883
IS - 5
ER -