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

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

A: Let G=GL(n,R) be the general linear group of degree n, that is, the group of all n×n invertible…

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

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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: 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…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

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: 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: 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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