On the converse of Gaschütz’ complement theorem

Abstract

Abstract Let 𝑁 be a normal subgroup of a finite group 𝐺. Let N ≤ H ≤ G N\leq H\leq G such that 𝑁 has a complement in 𝐻 and ( | N | , | G : H | ) = 1 (\lvert N\rvert,\lvert G:H\rvert)=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 \mathrm{Z}(N)\cap N^{\prime}\neq 1 . For metabelian groups 𝑁, the condition Z ⁢ ( N ) ∩ N ′ = 1 \mathrm{Z}(N)\cap N^{\prime}=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 ) \mathrm{Inn}(N) has a complement in Aut ⁢ ( N ) \mathrm{Aut}(N) .