The Nature Of Finite P-Group With All The Subgroups Of The Primary Abel P-Groups

The main work of this article is as follow:1 It has determined the types of the Finite p- Group with all the subgroups of the primary abel p- Group2 It has proven the nilpotency of the group which satisfies the condition through two methods. Also has proven the solubility of the group through the theorem.3 It has the following conclusion using the relationship between the order of elements and the number of conjugate classes s:1) If G is circulation group of order p~2, the group is a co ( p~2 -3) group .2)If G = p~3 ,exp(G)=p,the group is a co ( p~2 + p-3) group.3)If G is a primary abel p- Group , the group is a co ( pn-2) group.4 It has extraced the order and the number of the subgroup for each type of the group and drew the following conclusion:1)If G is circulation group of order p~2, G has only one subgroup of order p .2) If G = p~3 ,exp(G)=p,G has p~2 + p+1 subgroup of order p and only one Sylow p -subgroup of order p~2.3) If G is a primary abel p- group and G = pn (n≥1), -1≤m≤n,G has subgroup of order pm (1≤m≤n).