The subnormal structure of metanilpotent groups

Abstract

LetGbe a group with a normal nilpotent subgroupNsuch thatG/Nis periodic and nilpotent. IfG(p)/Nis the Sylowp-subgroup ofG/NandQ(p) is the maximalp-radicable subgroup ofN, it is shown thatGhas a bound on the subnormal indices of its subnormal subgroups if and only if there is a positive integercsuch thatG(p)/Q(p) is nilpotent of class at mostc, for all primesp. It is also shown that ifGis a periodic metanilpotent group andQis its maximal radicable abelian normal subgroup thenGhas a bound on its subnormal indices if and only if there is a positive integercsuch that for all primespthe Sylowp-subgroups ofG/Qare nilpotent of class at mostc.