By Bogert M.T., Chertcoff M.

**Read Online or Download A New Group of Dyes from Poison Gases through the 2-Aminothiazoles as Intermediates The Preparation of Thiazole Dyes of Doebner Violet Type PDF**

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**Additional info for A New Group of Dyes from Poison Gases through the 2-Aminothiazoles as Intermediates The Preparation of Thiazole Dyes of Doebner Violet Type**

**Sample text**

Cm , r1±v1 , . . , rk±vk , rk∓vk , . . , r1∓v1 ) = (r1±v1 , . . , rk±vk , cr1 , . . , crm , rk∓vk , . . , r1∓v1 ) = (cr1 , . . , crm ) . Thus EP can be viewed as a Z[G]-module. 9) are naturally isomorphic. The isomorphisms are deﬁed as follows. Let 1 m , . . , c±w ) ∈ EP , ci ∈ R, wi ∈ F. c = (c±w m 1 ±wk1 For a given rj ∈ R, j ∈ J, let w(c)j = ±wk1 . . ±wkl , where (rj is a subsequence of c with cki = rj . Deﬁne ±wkl , . . , rj ) ψP : EP → Z[G]⊕|R| by setting c → (ϕ∗ (w(c)1 ), . .

Then M (k) (G) = 0, k ≤ c. Proof. Let {gi }i∈I be elements in G such that {gi γc+1 (G)}i∈I is a basis of the free nilpotent group G/γc+1 (G). Consider the homomorphism f : F → G, where F is a free group with basis {fi }i∈I , given by setting fi → gi , i ∈ I. G/γc+1 (G) and an epimorThen f induces an isomorphism F/γc+1 (F ) phism (c) (c) M (c) (F )/ϕk (F ) → M (c) (G)/ϕk (G), k ≥ c + 1. 76 implies that f induces an isomorphism F/γn (F ) G/γn (G) for all n ≥ 1. 76 to get (c−1) M (c−1) (G) = ϕk (G), k ≥ c.

Proof. 48), we have N ∩ γm (G) = 0; hence, for any l ≥ m, there exists the natural isomorphism γl (G) γl (G), which immediately implies the assertion. 80 If G is a residually nilpotent group with M (n) (G) = 0 for all n ≥ 1, then G is an absolutely residually nilpotent. 14) that if, for a given group G, H1 (G) is torsion-free and H2 (G) = 0, then M (n) (G) = 0 for all n ≥ 1. 81 Let G be a group given by the following presentation: G = a, b, c | a = [c−1 , a][c, b] .