On self-normality and abnormality in the alternating group Ap

Abstract.

An old problem proposed by Huppert, Doerk and Hawkes motivates us to investigate the relationship between an abnormal subgroup and self-normalizing in non-solvable groups. A subgroup H of a group G is called second maximal if H is maximal in all maximal subgroups of G containing H. Our result is that if H is a second maximal subgroup of the alternating group A_p of prime degree, then H is abnormal in A_p if and only if H is self-normalizing.