On Groups All Of Whose Proper Subgroups Have Prime Power Order

In this paper, a further research is conducted to the finite solvable subsimple group, and a more detailed display of its nature and its structure is given. With the help of its structure, this paper points out that the non-commutative finite solvable subsimple group will necessarily be a qualitative element group. And referring to the nature of qualitative element group, it can be easily identified the conditions that the ordered prime divisors in a solvable initial finite subsimple group should meet. Suppose that G is always a finite group. The main conclusions are to be followed:Theorem1 Let G be a solvable subsimple group, G′is the only non-trivial normal subgroup of G , and G′is the primary group of the exchange of G ,and | G |= p~n q,|G′|=p~n,(p≠q,p,qprimenumber).Theorem2 Let G be a solvable subsimple group, if G is not Abelian, then all proper subgroups are p -group, that means if G = p~nq then every proper subgroup either group of order p n and of type {1 ,1,- ,1} or group of order q .Theorem3 G is a formal set p~n q of p -order subgroup of solvable simple group meeting, P is Sylowp -subgroup of G .So G = PQ,Q for which q -order cyclic group, P - G, P and G is the Frobenius for the Frobenius group of nuclear. Further,G have the main factor q, p~n, n is the index of p (modq).For a broader usage of theorem 2, a further research is conducted to any infinite group as long as its proper subgroup is p -. That is, how to identify the inner construction of IP -group in an easier way, when IP -group is non- p -group with the theories of∑- group and Hall's. And then with the help of the IP -groups' constructions, studies will be centered on its nature, and its relations with the finite solvable subsimple group and the finite groups in which every element has prime order and applying the two principles on IP -groups. The future studies can be on the wider use of IP -group in looser situations. The methods applied in the study of this paper include reversed proof, analysis, minimal reversed proof, etc. By studying the IP -groups, here come the following main results:Theorem4 Let G be a IP -group, then: (1)G is p -group. (2) G = pq( p, qare distinct primes). (3) G = p~nq( n > 1,p,qare distinct primes) and G is subsimple group, Sylowp -subgroup is the only nontrivial normal subgroup of G and is the Abelian group of order p n and type {1 ,1,- ,1,withnunits}.Theorem5 G is IP -group if and only if two proper subgroups which orders are prime and product is less than G are not commute.Theorem6 Let G be a IP -group, H < G, K