Solvable groups having system normalizers of prime order

Abstract

Let G be a solvable group having system normalizer D of prime order. If G has all Sylow groups abelian then we prove that l ( G ) = l ( C G ( D ) ) + 2 l(G) = l({C_G}(D)) + 2 , provided l ( G ) ≥ 3 l(G) \geq 3 (here l ( H ) l(H) denotes the nilpotent length of the solvable group H). We conjecture that the above result is true without the condition on abelian Sylow subgroups. Other special cases of the conjecture are handled.