3. Let GL2(R) be the group of all 2 x 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}. (a) (54) Prove that SL2(Z) is a subgroup of GL2(R). (b) (103) Let n be a positive integer, and let H be the subset of SL2(Z) given by 1+ np 1+ ns ng H = { A € SL2(Z) | A = for some p, q, r, sEZ nr Prove that H is a normal subgroup of SL2(Z). (주의: H가 subgroup of SL2(Z)이고 normal in SL2(Z)임을 모두 보여야 한다.)

3. Let GL2(R) be the group of all 2 x 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}. (a) (54) Prove that SL2(Z) is a subgroup of GL2(R). (b) (103) Let n be a positive integer, and let H be the subset of SL2(Z) given by 1+ np 1+ ns ng H = { A € SL2(Z) | A = for some p, q, r, sEZ nr Prove that H is a normal subgroup of SL2(Z). (주의: H가 subgroup of SL2(Z)이고 normal in SL2(Z)임을 모두 보여야 한다.)

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

3. 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}.
(a) (54) Prove that SL2(Z) is a subgroup of GL2(R).
(b) (103) Let n be a positive integer, and let H be the subset of SL2(Z) given by
1+ np
ng
H = { A € SL2(Z) | A =
for some p, q, r, s
1+ ns
nr
Prove that H is a normal subgroup of SL2(Z).
(주의 : H가 subgroup of SL2(Z)이고 normal in SL2(꾜)임을 모두 보여야 한다.)

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