Generalization And Application Of Tate’s Theorem

Based on research of so called transfer image TG(H) which was defined by Gagola and Isaacs in 2008 for the transfer homomorphism from a finite group G to its subgroup H. We prove that if H is a nilpotent Hall Ï€-subgroup of G then TG(H)∩O*(G)=[H∩OÏ€(G),H]. This result enrich the basic diagram in the transfer theory. Meanwhile we offer a more simpler group-theoretic proof, also provide a more general form of the Tate’s theorem, then obtain the direct product form of H/AÏ€(H), even prove that G/AÏ€(G) is isomorphic to a direct factor of H/AÏ€(H).The main conclusions of this paper as follows: Theorem 1. Let G be a finite group and Ï€ be a set of primes. Suppose that H is a nilpotent Hall 7r-subgroup of G, thenTG(H)∩OT(G)= [HnOÏ€(G),H].where TG(H) to denote the transfer image from G to H.Using theorem 1 we can deduce the following useful corollaries: Corollary 1. Let G be a finite group and p be a prime. Suppose that P is a Sylow p-subgroup of G, thenTG(P) n OP(G)= [P∩Op(G), P].Corollary 2. Let G be a finite group and Ï€ be a set of primes. Suppose that H is a nilpotent Hall 7r-subgroup of G, then the following are equivalent:{1)OÏ€(H)=H∩OÏ€(G).(2) AÏ€(H)=H∩AÏ€(G).(3) EÏ€(H)=H∩EÏ€(G).In the second part of this thesis, we give a direct product decomposition of the largest abelian Ï€ factor group. Theorem 2. Let G be a finite group and Ï€ be a set of primes. Suppose that H is a Hall ε-subgroup of G, write v:Gâ†'H/A"(H) to be the transfer homomorphism, thenUsing theorem 2 we can deduce the third useful corollary: Corollary 3. Let G be a finite group and Ï€ be a set of primes. Suppose that H is a Hall Ï€-subgroup of G, then G/AÏ€(G) is isomorphic to a direct factor of H/AÏ€(H).