2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H and k E K. Show that equivalence relation on G. ~ is an

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…

A: The solution is given as

Q: 9.12 Let G be a group and for elements a, b e G let a R b mean that there exists an element x = G…

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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: 3. Let G be a group and let SG be the set of all permutations on G. For each g € G, define Kg: GG as…

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Q: Let f:G to H be any group homomorphism, that is, f(x)f(y) is always f(xy). It will follow that also…

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Q: 7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E…

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Q: 2. First Isomorphism Theorem. Let G be a group and H-((.s)|ge G). (a) (b) (1) G is abelian. (i) H is…

A: Given G is a group then, G×G is a group of the type such that G×G={(a,b); where a, b belong to G}…

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: be a function from G to Sx, and for g E G and x E X, let g · x = a(g)(x). Prove that this defines a…

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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: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: (i.) Let F(x)= x². F is an automorphism on (R+,.) (j.) The set of non-zero real numbers with…

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Q: Let G be a group. Define a relation on G as follows: N exists an a EG such that y = axa. (In this…

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Q: 2 20. Let G be a group with subgroup H. Define a relation on G as follows: a b if b-¹a EH. Prove…

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Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

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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: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: 5. Prove (using the subgroup condition rather than using the axioms) that the set a b H = {( 6 ) :…

A: