Let G = GL2(R). Prove that each of the following subsets is a subgroup of G. {[: a (а) Н a, b e R, a # 0 0 а-1 (b) K = {A € G| det(A) E {±1}}. 1 1 1 0 1 1 A (c) L =

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

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Q: Prove or disprove that C* has a subgroup isomorphic to Z2 ⨁ Z2.

A: Given: Set, Z2⊕Z2We need to prove or disprove that C* has a subgroup isomorphic to Z2⊕Z2.

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Q: Here's a trickier example of a subgroup of GL(2, R): sin o 6)}~{( *)} cos d o sin cos o K= sin cos…

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Q: Let G = GL2(R). {[ {[ а (a) Prove that T = | a, c, d e R, ad is a subgroup of G. d a (b) Prove that…

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Q: (4) Let N and H be subgroups of G and let N 4G. Prove that NH is normal in H.

A: Note: Since you have posted a multiple question according to guidelines I will solve first…

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Q: Let C2 = {±1} be a subgroup of R". Prove that R = R+ x C2 where R is the set of real numbers.

A: Please check step 2 and 3 for the solution.!

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A: Let's define S1 as the set { (x,y)∣ x2+y2 = 1 } ⊂ R2 we may think of S3 as S3={ (a,b) ∈ C2:…

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Q: 3. Solve differential equation. Write clean, and clear. Show steps of calculations. (-xy sin x + 2y…

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Q: x = {[8 -d]|a,b€R} Prove that X*: =X\ {[%]} is a subgroup of GL(2, R).

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Q: Suppose  φ: ℤ x ℤ→ℤ defined by φ(m, n) = m - n. a. Show that φ is a homomorphism b. Compute Ker φ…

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Q: Compute the indicated values for the indicated homomorphisms.…

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