[Article] The Effects of an Homologous Series of Amines on by Loeb L. B.

By Loeb L. B.

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Additional info for [Article] The Effects of an Homologous Series of Amines on the Mobilities of Ions in Hydrogen Gas

Example text

To define j',, let xH be a coset of H . (x). This value is independent of the choice of coset representative x, and it is then trivially verified that j, is a homomorphism, is injective, and is the unique homomorphism satisfying our requirements. We shall say that j, is induced by [. can be factored into the following succession of homomorphisms : G s. G/H ~ 1m!.!... G'. Here, j is the inclusion of Im j" in G'. (ii) Let G be a group and H a subgroup. Let N be the intersection of all normal subgroups containing H.

Proof Let Fab(M) be the free abelian group generated by M. We denote the generator of Fab(M) corresponding to an element X E M by [x]. Let B be the subgroup generated by all elements of type [x + y] - [x] - [y] 40 I, §7 GROUPS where x, y EM. We let K(M) = Fab(M)/B , and let y : M --. K(M) be the map obtained by composing the injection of Minto Fab(M) given by x 1-+ [x], and the canonical map Fab(M) --. Fab(M)/B . It is then clear that y is a homomorphism, and satisfies the desired universal property.

The sum on the right is taken over the other orbits, and each index (G: Gx ) is then> 1, hence divisible by p. Since p divides the order of G, it follows that p divides the order of Z, hence in particular that G has a non-trivial center. Let a be an element of order p in Z, and let H be the cyclic group generated bya. Since H is contained in Z, it is normal. Letj" : G ~ G/H be the canonical map . Let pn be the highest power of p dividing (G: I). Then pn-l divides the order of G/H. Let K ' be a p-Sylow subgroup of G/H (by induction) and let K = f - l(K').

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