On self-normality and abnormality in the alternating group \(A_p\) (Q1305325)

A subgroup \(H\) of a group \(G\) is called second maximal if \(H\) is maximal in all maximal subgroups of \(G\) containing \(H\). It is proved here that for every second maximal subgroup \(H\) in the alternating group \(A_p\) of prime degree \(p\geq 5\) the following holds: \(H\) is an abnormal subgroup of \(A_p\) if and only if \(H\) is self-normalizing. This result should be contrasted with an example due to \textit{A. Feldman} [Arch. Math. 70, No. 1, 9-10 (1998; Zbl 0902.20007)] giving a self-normalizing second maximal subgroup in the unitary group \(U_3(3)\) that is not abnormal.