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

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Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

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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 groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2…

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Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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Q: Let G be a and group Z(G)=< g€G•xg=gx, VX€G} %3D be the center of G. Show that G is commutative if…

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Q: For any group G, Z(G) ≤ [G, G]. Prove or give a counter example

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Q: Theorem Let f: G H be a group homomorphism. Then, Im f≤ H.

A: Let us consider the mapping f:G→H . Then f is group homomorphism if  f(x·y)=f(x)·f(y) where, x,y∈G.…

Q: 2.2 Let f: → be defined by f(x) = 3x - 3. Prove or disprove that f is an isomorphism from the…

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Q: Let G be a group and let a be an element of G. Prove that a func- tion fa : G → G defined by fa(g) =…

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Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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Q: Prove that the trivial G is faithful if and only if G = {1G} representation of a nite group

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Q: Prove that if x is a group element then |x| = |x-1|.

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

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Q: Prove that if H is cyclic, then ø[H] is also cyclic.

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

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Q: let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

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Q: Exercise 8.6. Let G be a group. (a) Prove that G = {e} ≈ G. (b) Prove that G/{e} ≈ G. (c) Prove that…

A: 8.6 Let G be a group   (a) To Prove: G⊕e≅G (b) To Prove: G/e≅G (c) To Prove: G×e≅G

Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let G be a group, and let a E G.  Prove that C(a) = C(a-1).

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Q: Let G be a group. Show that for all a,b,c G, (acb-1)-1 = bc-la.

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Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

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Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

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Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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Q: Let a and b belong to a group G. Find an x in G such that xabx-1 = ba.

A: a and b belong to a group G. We need to find an X in such that xabx-1 = ba.

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: Let G, H be cyclic groups of order 2 and 3, respectively. Find the orders of all elements of G x H

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Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

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Q: Let (G, *), (G', *' and (G", ") be groups, and let :G G' and : G' → G" be isomorphisms. Prove that t…

A: We will use the definition of isomorphism to prove that the composition of two isomorphisms is again…

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 = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

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Q: Determine whether the set G is a group under the operation * G= integers Z ; a*b= a-b

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Q: (b) Given two groups (G,) and (H, +). Suppose that is a homomorphism of G onto H. For BH and A:{g…

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

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Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

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Q: 1 c = |G| gEG

A: The centralizer of a group G is the set of all elements in G that commutes with a particular element…

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…

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Q: Prove that the only action of the group G = C101 on a set X with |X| = 100 is the trivial action…

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Q: Let G be a group. Prove that Z(G) < G.

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Q: 2* Let f G H be a group homomorphism. Prove that if x E G and n is a natural number then f(x)= f(x)"

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Q: Let G be an infinite cyclic group. Prove that G (Z,+)

A: To show that any infinite cyclic group is isomorphic to the additive group of integers

Q: Let (G1, ) and (G2, *) be two groups and p: G1 G2 be an isomorphism. The

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Q: Let o be an automorphism of a group G. Prove that H = {x E G | $(x) = x} is a subgroup of G.

A: One step subgroup test: Let G,∘ be a group, then H,∘ is a subgroup of G,∘ if and only if, 1) H is…

Q: Prove that: Theorem 3: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if…

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Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

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