TY - JOUR

T1 - A pronormality criterion for supplements to abelian normal subgroups

AU - Kondrat’ev, A. S.

AU - Maslova, N. V.

AU - Revin, D. O.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

KW - complement of a subgroup

KW - finite simple group

KW - pronormal subgroup

KW - subgroup of odd index

KW - supplement of a subgroup

KW - FINITE SIMPLE-GROUPS

KW - ODD INDEX

UR - http://www.scopus.com/inward/record.url?scp=85018776312&partnerID=8YFLogxK

U2 - 10.1134/S0081543817020134

DO - 10.1134/S0081543817020134

M3 - Article

AN - SCOPUS:85018776312

VL - 296

SP - 145

EP - 150

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - 1

ER -