July  2023

  Τετάρτη 19/7  : B. Sambale (Leibniz University Hannover) 

         Τίτλος : Converses of Gaschütz' complement theorem

       Abstract: Let G be a finite group with an abelian normal subgroup N such that N\le H\le G and |N| is coprime to |G:H|.       A well-known theorem by Gaschütz states that if N has a complement in H, then N also has a complement in G. In contrast to the Schur-Zassenhaus theorem, Gaschütz' theorem does not hold when N is non-abelian. We show however that the theorem still holds when all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples whenever N'\cap Z(N)\ne 1.