Answered: Lagrange’s theorem Let G be a group…

Lagrange’s theorem Let G be a group offinite order and let H be a subgroup of G. Then the order of H divides’ the order of G. That is, I H I divides I GI

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter2: Basic Linear Algebra

Section2.5: The Inverse Of A Matrix

Problem 10P

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Lagrange’s theorem Let G be a group offinite order and let H be a subgroup
of G. Then the order of H divides’ the order of G. That is, I H I divides I GI.

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Let G be a (not necessarily finite) group with two subgroups H and K such that K ! H !G. The purpose of this question is to establish the index formula|G : K| = |G : H|·|H : K|.Let T be a transversal to K in H and U be a transversal to H in G.(a) By considering the coset Hg or otherwise, show that if g is an element of G, thenKg = Ktu for some t ∈ T and some u ∈ U.(b) If t, t! ∈ T and u, u! ∈ U with Ktu = Kt!u!, first show that Hu = Hu! and deduceu = u!, and then show that t = t!.(c) Deduce that TU = {tu | t ∈ T, u ∈ U } is a transversal to K in G and that|G : K| = |G : H|·|H : K|.(d) Show that this formula follows immediately from Lagrange’s Theorem if G is a finite group.
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