The Classification Of The 3-groups Of Class 3 All Of Whose Proper Subgroups Are Of Class≤2

If the nilpotency class of a finite p-group G is great than 2 and the nilpotency class of all its proper subgroups are at most 2,then G is called a minimal non-P2 group.The minimal non-P2 groups were classified for p>5 and p = 2.In this paper,we deal with the case of p = 3.If G is a minimal non-P2 group,then d(G)≤ 3.If d(G)= 2,then the minimal non-P2 groups are classified.If d(G)= 3,then an equivalent condition for a finite p-group to be a minimal non-P2 group is given.