### UNFORMATTED ATTACHMENT PREVIEW

Assignment 6 中国标准时间) Due: Saturday, August 14, 2021 11:59 am ( Assignment description Justify all claims in your solutions and state the results that you use. You may only use results that have been covered in Weeks 1-11. Submit your assignment Help After you have completed the assignment, please save, scan, or take photos of your work and upload your files to the questions below. Crowdmark accepts PDF, JPG, and PNG file formats. Q1 (10 points) Let 𝑀 be the matrix ⎡ 36 𝑀 = ⎢ ⎢ −36 ⎣ 24 Find matrices 𝑃 , 𝑄 ∈ GL 3(ℤ) −24 72 −36 54 12 ⎤ ⎥ ⎥ . −24 ⎦ such that 𝑃 𝑀 𝑄 is in Smith normal form. You must show the steps you took to find the matrices 𝑃 and 𝑄. Recall that GL𝑛(ℤ) is the set Mat𝑛×𝑛(ℤ)× of invertible elements of Mat𝑛×𝑛(ℤ) , that is, GL𝑛(ℤ) is the set of matrices 𝑀 ∈ Mat𝑛×𝑛(ℤ) for which there exists a matrix 𝑀 ′ ∈ Mat𝑛×𝑛(ℤ) such that 𝑀 𝑀 ′ ′ = 𝐼𝑛 = 𝑀 𝑀 . It can be shown that GL 𝑛(ℤ) = {𝑀 ∈ Mat𝑛×𝑛(ℤ) : det(𝑀 ) ∈ ℤ × = {±1}}. Time left Show Q2 (10 points) Some exercises on the Week 10 exercise sheet will give you all the tools you need for this problem. You should work through those exercises, but you can use them here without including proofs of them. Let 𝑏1 = (2, 2) 1. Is 𝛽0 and 𝑏2 = {𝑏1, 𝑏2} 2. Let 𝐵 = span ℤ 3. Find a basis 𝛽 and = (−4, 4) a basis of ℤ2 ? . Prove that 𝐵 is a free abelian group of rank 2. (𝛽0 ) ′ 1 ′ 2 = {𝑏 , 𝑏 } ′ 𝑏 = 𝑑2𝑐2 2 . of 𝐵, a basis 𝛾 = {𝑐1, 𝑐2} of ℤ2 , and positive integers 𝑑1 ∣ 𝑑2 such that 𝑏′1 = 𝑑1𝑐1 . 4. Express ℤ2/𝐵 as a direct product of cyclic groups. Q3 (10 points) Let 𝐴 be an abelian group and let 𝐵 be a free abelian group of finite rank. Prove that for every surjective homomorphism 𝜋 : 𝐴 → 𝐵 there exists a homomorphism 𝜄 : 𝐵 → 𝐴 such that 𝜋 ∘ 𝜄 = id𝐵 . Q4 (10 points) Let 𝑝 be a prime and let 𝔽𝑝 = ℤ/𝑝ℤ Let 𝑃 1(𝔽𝑝) be the set of all lines ℓ 1. For each 𝐴 SL2 (𝔽𝑝) ∈ SL2 (𝔽𝑝) on 𝑃 1 (𝔽𝑝) 𝔽𝑝 ⊆ 𝔽 and ℓ 2 𝑝 ∈ 𝑃 through the origin of 𝔽𝑝2 . 1 (𝔽𝑝) , define 𝐴 ⋅ ℓ = {𝐴𝑣 : 𝑣 ∈ ℓ} . Prove that this defines an action of . 2. Prove that |𝑃 1(𝔽𝑝)| span . Note that SL2 (𝔽𝑝) acts on 𝔽𝑝2 via matrix multiplication. = 𝑝 + 1 (𝑣 1), … , span 𝔽𝑝 . (Hint: find 𝑝 + 1 vectors 𝑣1, … , 𝑣𝑝+1 ∈ (𝑣 𝑝+1) 𝔽 2 𝑝 such that the elements of 𝑃 1(𝔽𝑝) are .) 3. Define Time left PSL2(𝔽𝑝) := SL2 (𝔽𝑝)/𝑍 Show where 2 𝑍 = {𝑘𝐼2 : 𝑘 ∈ 𝔽𝑝} ∩ SL2 (𝔽𝑝) = {𝑘𝐼2 : 𝑘 ∈ 𝔽𝑝, 𝑘 = 1}. Using the action of SL2 (𝔽𝑝) on 𝑃 1(𝔽𝑝) from Part 1, construct an injective homomorphism from PSL2(𝔽𝑝) to the symmetric group on 𝑃 1(𝔽𝑝) . 4. Using the injective homomorphism from Part 3, prove that PSL2(𝔽2) ≅ 𝑆3 , PSL2(𝔽3) ≅ 𝐴4 . (You can use the fact that 𝑆4 has a unique subgroup of index 2 without proving it.) Time left Show QQ . A组对接员蓝天· 写手在 Q1 (10 points) Let G be a finite cyclic group of order 60. let H be a subgroup of G of order 12, and let k be a subgroup of G of order 30. How many subgroups of G are contained in H, but not in K? (Justify your answer.) + Drag and drop an image or PDF file or click to browse… Q2 (10 points) Let G be a finite group. For each prime p, define H, = {8EG:p + 0(g)). 1. Prove that if G is abelian, then H, 3 G for every prime p. 2. Prove that if G = S3, then Hy is not a subgroup of G. + Drag and drop an image or PDF file or click to browse. | 文 … … ] O Q3 (10 points) Lata E So be the element 1 2 3 4 5 6 7 8 9 (6 7 3 8 5 9 4 2 1 1. Prove that there does not exist an integer k such that a = 755 (Hint: You can prove this without simplifying ok or 055) 2. Prove that there do not exist x, y E S, such that o = xyxy + Drag and drop an image or PDF file or click to browse.. Q4 (10 points) Click the following link to download a table that is necessary for this question: TT1 Cayley_Table.pdf. The table that you downloaded is a Cayley table of a group of order 12, but with most its entries deleted. The element denoted by e is not assumed to be the identity element 1. Prove that is the identity element of the group. (Hint: Look at the first row) 2. Prove that e = c?. (Hint: Use the rows corresponding to a and c.) 3. Prove that b = a (Hint: Use ca = a) Conclude that b is the identity element of the group. 4. Compute the orders of a and b 5. Is G,D,? 查看原图 (212K) 4 文 … … 3. FIUVUD 4. Compute the orders of a and b. 5. Is GED? + Drag and drop an image or PDF file or click to browse… Q5 (10 points) Let G be a group and let A be an abelian group. Let Hom(G, A) be the set of all homomorphisms from G to A. For each 4.7 € Hom(G,A), define .y: G → A by (0-w)(g) = $(g)v(g) for all g € G. Prove that .V e Hom(G, A) for all , V E Hom(G,A), and that Hom(G, A) is an abelian group under the binary operation Hom(G, A) Hom(G, A) Hom(G, A) that maps (0.7) to .v. + Drag and drop an image or PDF file or click to browse… Before doubt | 文 … … ] O Q1 (10 points) Is every subgroup of Q8 X Qnormal? + Drag and drop an image or PDF file or click to browse… Q2 (10 points) Let G be the subgroup of GL4(R) defined by ={(6) A, BE GL2(R) and * E Mat2x2 (R) »} Let N be the subgroup of G defined by * E Mat2x2(R) where 12 E GL2(R) is the 2 x 2 identity matrix. Prove that N is a normal subgroup of G and give an isomorphism between G/N and a group that is not a quotient group. + Drag and drop an image or PDF file or click to browse… Q3 (10 points) Let G be a group. Suppose that the quotient of G by one of its abelian normal subgroups is abelian. Prove that if H is a subgroup of G, then the quotient of H by one of its abelian normal subgroups is abelian. (Hint: Apply the Second Isomorphism Theorem.) + Drag and drop an image or PDF file or click to browse… Q4 (10 points) Let p, q, r be distinct primes and let A be a finite abelian group of order pqr. Without using the classification of finite abelian groups, prove that A = Z/pqrZ. (Hint: Show that A ZipZ X Ziqz x ZİrZ.) + Drag and drop an image or PDF file or click to browse… Q5 (10 points) Let A be a finite abelian group. Without using the result that for finite groups G1, … ,Gn we have exp(G1 X … X Gn) = lcm(exp(G1), …, exp(Gn)), prove that exp(A) is the largest invariant factor of A. Conclude that A has an element of order d for all positive integers d | exp(A). Drag and drop an image or PDF file or click to browse…

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