Let GL2(R) be the group of all 2×2 nonsingular matrices with entries in R, and let SL2(Z)be the subset of GL2(R) given bySL2(Z) = {A ∈ GL2(R) | all entries of A are in Z and det(A) = 1} . Let n be a positive integer, and let H be the subset of SL2(Z) given byforH = { A € SL2(Z) | A :1+ npnqfor some p, q, r, s € Znr1+ ns Prove that H is a normal subgroup of SL2(Z).

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Answered: Let n be a positive integer, and let H…

Let n be a positive integer, and let H be the subset of SL2(Z) given by -frcs 1+ np nq H A E SL2(Z) | A = for some p, q, r, s E Z nr 1+ ns

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let GL2(R) be the group of all 2×2 nonsingular matrices with entries in R, and let SL2(Z)
be the subset of GL2(R) given by
SL2(Z) = {A ∈ GL2(R) | all entries of A are in Z and det(A) = 1} .

Let n be a positive integer, and let H be the subset of SL2(Z) given by
for
H = { A € SL2(Z) | A :
1+ np
nq
for some p, q, r, s € Z
nr
1+ ns

Prove that H is a normal subgroup of SL2(Z).

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