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: Use Cayley's theorem to find an explicit subgroup of Sg which is isomorphic to the quaternion group…

A: Given Information:Group and Quaternion group .To find:The subgroup of  isomorphic to the Quaternion…

Q: Suppose F1 and F2 are distinct reflections in a dihedral group Dn.Prove that F1F2 ≠ R0.

A: Given: F1 and F2 are distinct reflections in a dihedral group Dn. To Prove: F1F2≠R0.

Q: Let H be a subgroup of R under addition. Let K = {2ª | a E H}. Prove that K is a subgroup of R*…

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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: Suppose o is a homomorphism of groups fom G onto H. If H has an element of order 100, then show that…

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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.

Q: Find a proper subgroup of the group of integers Z under addition and prove that this subgroup is…

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Prove that in F7 the cyclic subgroup generated by x is a normalsubgroup.

A: Given: F7We need to prove that the cyclic subgroup generated by x in F7 is a normal subgroup.

Q: Let H = {a + b√3 | a, b ∈ Q and a and b are not both zero}. Prove or disprove: H is a subgroup of…

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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: 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: 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: 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…

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: 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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Q: Find the partition of Ze into cosets of the subgroup H = {0,3}.

A: In this problem we have to find the partition of  into cosets of the subgroup .

Q: Prove that in F7 the cyclic subgroup generated by x is a normalsubgroup.

A: Given : F7  is the cyclic subgroup generated by x. To Prove That: F7 is a normal subgroup.

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: Show that A4 is nolmal by finding a homomorphism whose kernel is A4.

A: If (alternating group) An is the normal subgroup of Sn, then by first isomorphism theorem implies…