The Application Of Schur-Zassenhaus Theorem In Interior G-Algebra

Let O be a complete discrete valuation ring, k is the residue field of O, the characteristic of κ is the prime ρ and κ is algebraically closed. Let G be a finite group, Abe an interior G-algebra over O, Rδ be a pointed group on A. Choose l∈δ, let Aδ=lAl, then Aδ is an interior R-algebra. Denote by Aδ*, AδR, Rl and J(AδR) the multiplicative group of all invertible elements of As, the set of all R-fixed elements in Aδ, the image of R in Aδ and the Jacobson radical of AδR. Denote by NAδ*(R) the normalizer of Rl in Aδ*, by FA(Rδ) the A-fusion group from Rδ to Rδ. In this paper, we prove the following theorem:Theorem: If R is a p-group, FA(Rδ) is a p’-group, and the map R�'Aδ*,u�'ul is injective, then the subgroup (Rl)(l+J(AδR)) has a complement X in NAδ*(R) containing κ*and any such two oomplements are conjugate by an element of (Rl)(l+J(AδR)).