Character Of ¦Ð-blocks Of Nature

Suppose that G is a finit group, andÏ€is a set of prime number. Let B (?) Irr(G) be non-empty, then B is called aÏ€- block iff for every p∈π, B is a union of p-blocks and B is minimal relatively.In this paper, we will compare the peroties ofÏ€-block's with p-block's, and get following conlusions:Theorem 1 If p(?)|G| for any p∈π, then IBr(G) = Irr(G).Theorem 2 Setθ(x) =(?),θ(x) =θ(x_Ï€'),x∈G.Ifθ∈Z[Irr(G)]∪Z[I Br(G)],thenθandθare generalized characters of G.Theorem 3 Set a^p = v^p(|G|), then p^a(p)|Φ_j(1), for any p∈π,j∈L.Theorem 4 Set H be aÏ€'-subgroup of G, thenΦ₁ is an irreducible composition ofcharacter (1_H)^G. If H is aÏ€-complement of G, thenΦ₁ = (1_H)^G.Theorem 5 For k∈K and b^p = v^p(χk(1)),1/Π_p∈πp^b(p)χk is a generalized character ofG, while (?)_q∈π,1/q 1/Π_p∈πp^b(p)χk is not.Theorem 6 D(B) is not of the form (?) for any B∈Blk_Ï€(G).Theorem 7 For decomposition matrix D and Cartan matrix C, if there is a matrix Psuch that PD = (?), where D₁ is aλ×λinvertible matrix, P₁ is a matrix. Thenthere exists a k×k matrix S such that SD = (?) and D^TS^TSD = C.Theorem 8 For decomposition matrix D and Cartan matrix C, if there is a matrix P suchthat PD =(?), where D₁ is aλ×λinvertible matrix, P₁ is a matrix such thatP₁^TP₁= 0 and P₁P₁^T= 0. Then there exists a k×k matrix S such that SD = (?)and D^TS^TSD = C.Theorem 9 For composition matrix D and Cartan matrix C, if k =λ+ 1, k= 2ⁿ withn∈N, then there exists a k×k matrix S such that SD =(?), where D₁ isλ×λ invertible matrix, and D^TS^TSD = C.Theorem 10 Det(C) =Π_j∈λ|C_G(x_j)|_Ï€.Theorem 11 If B∈Blk_Ï€(G), k∈X, then KerB =O_Ï€'(Kerχk).Theorem 12 If B∈Blk_Ï€(G), then KerB =∩_j∈L(B)Ker(?)j.Theorem 13 Suppose that N(?)G,θ∈I Br(N),(?)∈IBr(G) is an irreducible compositionofθ^G, andθ=θ₁,θ₂,…,θ_t is distinct conjugates ofθin G, then (Φ_?)_N = e(?)Φ_θi,where e = I((?)N,θ).