A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and Hg are conjugate in the subgroup <H,Hg>. We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = NU(H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp6n(q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.
|Журнал||Proceedings of the Steklov Institute of Mathematics|
|Состояние||Опубликовано - 1 апр 2017|