Central Expansion Of Degree 3 Of The 3-groups Of Maximal Class And Its Applications

Let N and H be arbitary groups.If there exists a group G which has a normal sub-group N ≤ Z(G)such that N ≌ N and G/N ≌ H,then G is called a central extension of N by H.In this paper,it is given that the classification about the central extension of the 3-group of maximal class with an abelian maximal subgroup.Based on the classi-fication,an equivalent condition for a finite 3-group to be of maximal class are obtained.In addition,a simple characterization for a finite p-group to be of maximal class are also given for p ≥ 5.