Let G1 and G2 be any two groups and θ : G1 → G2 be a group isomorphism. Let H1 ≤ G1. Prove that H2 = θ(H1) ≤ G2 and |G1 : H1| = |G2 : H2|

Q: Let (G1, ) and (G2, *) be two groups and p: G1- G2 be an isomorphism. Then * G2 is abelian if G1 is…

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Q: Exercise 8.3. (a) If H₁ and H₂ are subgroups of groups G₁ and G₂, respectively, prove that H₁ÐH₂ ≤…

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Q: Let (G1, ) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then O G2 might be abelian…

A: G2 might not be abelian even if G1 is abelian.

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

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Q: that a and xax^-1 have same order for all x belongs G.

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Q: Let (G1, •) and (G2 , *) be two groups and :G1 G2 be an isomorphism. Then * G2 might be abelian even…

A: we know that,  If two groups are isomorphic, then all the property will be similar i.e. one  is…

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Q: Let G = (-1,1) and (x,y) --> x ○ y = x + y/1 + xy is a binary operation on G. Let R+ = (0, ∞), with…

A: Given: G,O  with G=(-1, 1) ,  (x, y)→ x O y = x+y1+xy   and ℝ+, ·, ℝ+ = (0, ∞) with standard…

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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A: This is related to isomorphism

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A: Solution: We know that the permutaion τ on a non empty set A is a bijective map from A to A. That…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^2= e a =…

A: Since f is a group homomorphism , where f(x)=a∗x∗a−1,    x∈G. So a^-1=a implies self inverse implies…

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Q: Let (G1, •) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then * G2 might be abelian…

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Q: . Let G be the additive group Rx R and H = {(x,x) : x E R} be a subgroup of G. Give a geometric…

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Q: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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Q: Let G be a finite group. The center Z(G) of G is defined to be the set of all group elements z E G…

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Q: Let f be the homomorphism f Z15 GL(2) of groups such that (1)-A, where A- a) Give the definition of…

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Q: Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b)…

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Q: Let(G,*) and (H,#) be a groups if f: G H and g: H G are homomorphism such that gof = IG.f og = IH…

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Q: Let f be the homomorphism f Z1s → GL(2) of groups such that (1) = A, where A = a) Give the…

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Q: Let f: - be defined by f(x) = 3x-3. Prove or disprove that f is an isomorphism from the additive…

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Q: Let (G1, •) and (G2 , *) be two groups and p : G1 – G2 be an isomorphism. Then * O G2 is finite if…

A: We will use basic results of group theory

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A: We will use property of homomorphism to solve the following question

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a = e O a^4…

A: Given that f:G→G be a function such that fx=axa

Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: Let G1 and G2 be any two groups and θ : G1 → G2 be a group isomorphism. Let H1 ≤ G1. Prove that H2…

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Q: Let G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

Q: Suppose that f: G → G such that f(x) = axa. Then fis a group homomorphism if and only if a^4 = e a^3…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider the two multiplicative groups (R.) and (R;. .) and let f: (R.) → (R.) be a map defined by…

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Q: Suppose that f: G G such that f(x) and only if = axa. Then f is a group homomorphism if -> a = e a^3…

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Q: Let G be a group of odd order. a) Prove that if x E G, then (x2) = (x). b)…

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Q: Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

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Q: Let G be a group and define the map o : G → G by $(g) = g¬1. Show that o is an automorphism if and…

A: Automorphism and abelian group

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Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: 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…

A: Given :  fa,b : ℝ→ℝ  defined by  fa,b(x) = ax+b  ∀ x∈ℝ.   To show :  (i) G = fa,b : a, b ∈ℝ and a≠0…

Q: Suppose that f:G → G such that f(x) = axa. Then f is a group homomorphism if and only if a^4 = e %3D…

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Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

Q: Let G be a group, and define N to be the cyclic group N = ⟨xyx−1y−1 | x,y ∈ G⟩. (a) Prove N is a…

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Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

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Q: Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove…

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Q: 1. Let G be any group. Show that the function f: GG defined by f(x) = x² is group homomorphism if…

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Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

A: Given:G and H be the groups.

Q: Let (G1, •) and (G2 , *) be two groups and p : G1→ G2 be an isomorphism. Then * G2 might not be…

A: G2 is abelian if G1 is abelian.