We show that each sporadic simple group G can be uniquely determined by the set of the orders of the maximal abelian subgroups of G.

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A subgroup H of a group is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H |) = 1. Using the concept of s-semipermutable subgroups, some new characterizations of p-nilpotent groups are obtained and several results are generalized.

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The structure of finite groups is widely used in various fields and has a great influence on various disciplines. The object of this article is to classify these groups G whose number of elements of maximal order of G is 20.