The Normality And Applications Of Thompson Subgroups

In 1964, J. Thompson introduced three different characteristic subgroups Jr(P), Je(P) and Jo(P) for an arbitrary finite p-group. Using these three subgroups, he proved similar Thompson p-nilpotence theorems. This result not only generalized correspond-ing theorems of Frobenius p-nilpotence group, but also had significant influence on the development of group theory. The original proof of Thompson’s theorem were too dif-ficult and complex. In 2008, I. M. Isaacs gave the sufficient conditions for Je(P) to be normal in G, i.e.Je(P)< G. As an application, it supplied a much simpler proof of Thompson p-nilpotence about Je(P).In this paper, we study the normality of the Thompson subgroup Jr(P) and give a similar normality theorem of Jr(P). In particular, our results generalize I. M. Isaacs’s theorem of Je(P).Theorem 1. Let P € Sylp(G), where G is a finite group. Assume the following conditions.(1) G is p-solvable;(2) p ≠ 2;(3) A Sylow 2-subgroup of G is abelian;(4) Op’(G)= 1;(5) P= CG(Z(P)).Then Jr(P)(?) G.As an application, we offer a simplified proof of Thompson’s p-nilpotence theorem on Jr(P).Corollary 1. Let P ∈ Sylp(G), where G is a finite group, p≠ 2, and assume that CG(Z(P)) and NG;(Jr(P)) have normal p-complements. Then G has a normal p-complement.From Corollary 1, we give a simple proof of Thompson’s p-nilpotence theorem on Je(P).Corollary 2. Let P ∈ Sylp(G), where G is a finite group and p≠2. Assume that CG(Z(P)) and NG(Je(P)) have normal p-complements. Then G has a normal p-complement.The second main result in this paper is the following theorem, which shows that in a finite group G, if the normalizer NG(P) of a Sylow P-subgroup P is p-nilpotent, then the subgroup CG(Z(P)) in Thompson’s theorem is actually equal to NG(Z(P)).Theorem 2. Let P € Sylp(G) is a Sylow p-subgroup of G, where G is a finite group, and p is a prime. If NG(P) is p-nilpotent, then NG(Z(P))= CG(Z(P)).An easy application of Theorem 2 is the following:Corollary 3. Let P ∈ Sylp(G) is a Sylow p-subgroup of G, where G is a finite group. If NG(J(P)) is p-nilpotent, where J(P) ∈{Jr(P), Je(P), JO(P)} is a Thompson subgroup, then NG(Z(P))= CG(Z(P)).