A nonabnormal subgroup contained only in self-normalizing subgroups in a finite group

Abstract.

If U is abnormal in G, then N _G (V) = V for every V≥U. The converse is known to be true for solvable groups, but in Finite Soluble Groups, Doerk and Hawkes indicate that its truth is not known for G finite and nonsolvable. In this note, we provide a counterexample. We prove:¶¶Theorem. In the unitary group G = U ₃ (3) there exists a nonabnormal subgroup U isomorphic to S ₄ such that U ≤ V implies N _G (V) =V.