The Classification Of Finite P-groups Which Has Abelian Maximal Subgroup Generated By Two Elements

The main purpose of this article is to classify the finite p-groups which have at least one abelian maximal subgroup generated by two elements. For the convenience of narration, we introduce a new concept of M_n-groups. Assume that G is a finite non-abelian p-group. If G has exchange abelian maximal subgroup, and the number of generators of an abelian maximal is at least n, then G is called M_n-groups. M₁-group has been classificed. This paper has classificed M₂-group completely, among them p is greater than 3.