1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under multiplication. ´cos no sin no -sin no COs no )loER, nE 2}. a. Let H == Show that H C M2(R).

Q: SUCH THAT LET H BE A PROPER SUBGROUP OF G V x,y € G-H, xy EH. PROVE THAT HAG.

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Q: Does Sn have a subgroup isomorphic to An other than An (n ≥ 5).

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Q: Let H = {A E GL(2, R) | det A is an integer power of 2}. Show that H is a subgroup of GL(2, R).

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Q: Prove that SL,(R) is a normal subgroup of GL,(R).

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

Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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Q: What can you say about a subgroup of D3 that contains R240 and areflection F? What can you say about…

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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.!

Q: For any positive integer m, show that every subgroup of Zm is an ideal of Zm-

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Q: Suppose there is a homomorphism Φ from G onto Z2 ⨁ Z2. Provethat G is the union of three proper…

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Q: Prove that any non-trivial subgroup of (Z, +) is isomorphic to (Z, +).

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Q: Q,+ ,+,*) is not a proper subgroup of R+ under multiplication? T or F

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Q: to a subgroup of Sn for n ≥ 3.

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Q: Consider the subset ℚ(sqrt3) = {a + b : a, b ∈ ℚ} of ℝ. Show that ℚ (sqrt3) is a subgroup of ℝ under…

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Q: There exists a one-one correspondance between a   subgroups H of G and a left coset aH of H in G.

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Q: Let G= { a+bie C: a and bare rational numbers, not both zero}. Prove that G with the operation of…

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Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

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Q: A={[₁]IXER,YER} | x R* А

A: A is not a subgroup of GL(2, ℝ) because it is not closed under matrix multiplication .

Q: Prove that a kernel of a homomorphism is a subgroup. 52.

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Q: qoustion

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Q: Question 3: a. Let (H,+) ≤ (R, +), let K = {2h : h E H}. Prove that (K,x) ≤ (R*,x). (Clarification:…

A: We shall use subgroup test to solve this problem

Q: 4. If p: G → G' is a homomorphism, prove that ker ý is a normal subgroup of G.

A: To prove that kernal is a normal subgroup of G

Q: def S ER a ca> Show thats is a subgroup ob (M2(R),+). (6) Show that ib A, B ES then ABES. (C) Find…

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Q: 3) Show that the subgroup of Dg is isomorphic to V4.

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Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Let H be a subgroup of G, and let a and b belong to G. Then |aH| = |bH|

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Q: If n be any given positive integer, show that the mapping f: Co→ Co defined by f(z) = z¹ is an…

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Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

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Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

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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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