The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superaigebra(L,[p]).We show first that if L=A⊕A⊕…⊕A,then φ(L)=φ(A)+φ(A)+…+φ(A),where each Ais a p-ideal of L. We then obtain two results: F(L)=φ(L)=J(L)=Lif andonly if L is nilpotent;F(L) and F(L) are nilpotent ideals of L if L is solvable. In addition, necessaryand sufficient conditions are found for χ-free restricted Lie superaigebras. Finally, we discuss therelationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.