  A subgroup of a finite group G is said to be second maximal if it is maximal
in every maximal subgroup of G that contains it. A question which has received
considerable attention asks: can every positive integer occur as the number of
the maximal subgroups that contain a given second maximal subgroup in some
finite group G? Various reduction arguments are available except when G is
almost simple. Following the classification of the finite simple groups, finite
almost simple groups fall into three categories: alternating and symmetric
groups, almost simple groups of Lie type, sporadic groups and automorphism
groups of sporadic groups. This thesis investigates the finite alternating and
symmetric groups, and finds that in such groups, except three well known
examples, no second maximal subgroup can be contained in more than 3 maximal
subgroups.
