Some Properties Of Restricted Lie Triple Systems

In this thesis, I give some basic conceptions and properties of restricted Lie triple systems and discuss the Erattini subsystems and p-solvable restricted Lie triple systems. Moreover, I obtain some sufficient and necessary conditions of p-solvable restricted Lie triple systems. The main results follow:Theorem 1 Let (T,[p]) be a restricted Lie triple system over F, then the following statements hold:1) If C is a p-subsystem of T and B is a p-ideal of T such that B (?) F(C), then B (?) F(T).2) If C is a p-subsystem of T and B is a p-ideal of T such that B (?)φ{C), then B (?)φ(T).3) If T is solvable, then J(T) = T¹ .4) If T is nilpotent, then each maximal subsystem H of T is its an ideal. Theorem 2 Let (T,[p]) be a restricted Lie triple system over F. If T-A₁⊕A₂⊕…⊕A_n, where each A_i is a p-ideal of T, thenφ_p(T) =φ_p(A₁) +φ_p(A₂) +…+φ_p){A_n).Theorem 3 Let (T,[p]) be a restricted Lie triple system over F. If T is p-solvable, then T must be solvable.Theorem 4 Let (T,[p]) be a restricted Lie triple system over F. If x^[pⁿ(x)] = x and 1 (?) n(x) (?)k for any x∈T, then T is abelian if and only if T is p-solvable.Theorem 5 Let (T,[p]) be a restricted Lie triple system over F. If x =Σα_i(x)x^[p]ⁿi(x) for all x∈T, whereα_i(x)∈F,n_i (:r)∈N, then T is abelian.Theorem 6 Let (T,[p]) be a restricted Lie triple system over F such that x =Σα_i(x)x^[p]ⁿi(x) for all x∈T, whereα(x)∈F,n_i(x)∈N. If A is an abelian ideal of T, then A(?)C(T).Theorem 7 Let (T,[p]) be a restricted Lie triple system over F. If I is a p-solvable ideal of T, and T/I is p-nilpotent, then T is p-solvable.