For a, b ∈ R with a 6= 0, define a map fa,b : R → R as fa,b(x) = ax + b for all x ∈ R. Let us define G = {fa,b : a, b ∈ R and a 6= 0}. Show that G is a group under composition operation. Further, show that the set N = {f1,b : b ∈ R} is a normal subgroup of G, and that G/N  congruent R∗ ,where R∗ = R \ {0} is the multiplicative group of non-zero real numbers.

Q: subgroups of G. Let x,y e G. We define the relation x ~ y by the following definition: x ~ y iff y =…

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Q: Let X be a set, G a group, Sx={f:X→X | f is bijective on G} is a group with composition. Given x0∈X…

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Q: 2. Let G = (R*, .), the multiplicative group of nonzero real numbers. Let H = {x ER: x -1} be the…

A: 2. Let , the multiplicative group of non-zero real numbers.Let  be the group with operation .Prove…

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: Let X be a set, G a group, Sx={f:X→X | f is bijective on G} is a group with composition. Given x0∈X…

A: Solution:- given G is a group and X is a subset of G . Here we have to Show subgroup property

Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

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Q: Let A = {fm,b : R → R | m ± 0 and fm,b(x) : = mx + b, m, b e R} be the group of affine m b functions…

A: We are given two groups.  A={fm,b:ℝ→ℝ|m≠0 and fm,b(x)=mx+b,m,b∈ℝ} B=mb01 | m,b∈ℝ,m≠0. We need to…

Q: Question 1 Consider the groups Z² = Z× Z and Z³ = ZxZxZ and the map ƒ : Z³ → Z² defined by the rule…

A: We know that Given groups G (with operation *) and H (with operation #), the direct product G × H is…

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: Let n > 2 be an integer, and let X C Sn × S, be the set X = {(ơ, T) | 0(1) = T(1)}. Show that X is…

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Q: 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a)…

A: 2a Given set is A=a00d:a>0 and d>0. We have prove that set A is the subgroup of GL2,R. For the…

Q: Show that the center Z(G) is a normal subgroup of the group G. Please explain in details and show…

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Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

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

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Q: SIZ): Set of all fiZ-→Z which are both one-one S(Z): on ti and S(Z) is a group under composition of…

A: Given, S(ℤ)=f|f:ℤ→ℤ, 1-1, onto is a group under composition of function. And H=fn:ℤ→ℤ|fn(x)=n+x.…

Q: Let G be a group and let H and K be normal subgroups such that Hnk= G/H x G/K be the map o(g) = (Hg,…

A: A mapping from a group (G,o) to a group (G',o') is called a homomorphismif f(aob)=f(a)o'f(b)where a,…

Q: Let a e G. Prove that $(a") = ¢(a)" for all n e Z.

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Q: 2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G ×…

A: Step 1: Step 2: Step 3: Step 4:

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: Find a matrix M such that M-¹0 (8) M = (19) for every element g € S3. Conclude that the…

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Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.