21. Let G be an abelian group with subgroup H. Let G/H be the set of cosets of H in G. Define multiplication of congruence classes by aH-bH = abH. Prove that if aH = a'H and bH = b'H, then abH = a'b'H, and so multiplication of cosets is well-defined. Prove that G/H is an abelian group with this multiplication. This is called the quotient group of G by H.

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A: We will prove this by using the concept of cyclic subgroups and the properties of abelian groups.…

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A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: Prove or Disprove: If (G, *) be an abelian group, then (G, *) a cyclic group?

A: If the given statement is true then we will  proof the statement otherwise disprove we taking the…

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Q: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: 7. Let (G, *) be a group and let a G. Define C(a)= {bEG: ab ba Z(G) = {bEG: ab-ba for all a € G}…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: Let G be an Abelian group. Determine all homomorphisms fromS3 to G.

A: Given: G is an Abelian group.To find: All homomorphisms from S3→G.

Q: Write the Caley table of each of the subgroups of each abelian group of order 8.

A:

Q: (4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹.…

A: Let  be  a  group  and  assume  that Suppose  that Since,  and  is  a  group, it follows  that…

Q: Determine the group Z42/ and find a group that it is isomorphic to it. Generalize your observation.

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Q: 6. Show that (an)neN C R is bounded if and only if lim supn→∞ an < +∞o.

A: To Prove:  ann is bounded sequence if and only if limsupan&lt;+∞.

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

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Q: Define the concept of isomorphism of groups. Is (Z4,+4) (G,.), where G={1,-1.i.-i}? Explain your…

A: Lets solve the question.

Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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Q: 8. Let G be a finite abelian group. (a) Let d be a positive divisor of |G|. Prove that G has a…

A: a) We need to prove that if d is a positive divisor of |G| then G has a subgroup of order d. Let the…

Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: 6. Let G be GL(2, R), the general linear group of order 2 over R under multiplication. List the…

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Q: If (G, * ) is a group with a a for all a in G then G is abelian

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Q: 4) Let set G be the Cartisian product of G1 × G2, where G1 and G2 are 2 groups. Define a binary…

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Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: 5. Let G be a group and define z" z * *** *z for n factors c, where z E G and n E Z+. (a) Suppose…

A: Given that G is a group. Also zn is defined as follows. zn=z*z*z*...*z (a) Use mathematical…

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: If (G, *) is a group, such that (a * b)² = a? * b² va, be G, then G is alan %3D إختر أحد الخيارات…

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Q: ^^)^^\Y) 13. If * is defined on Z such that a*b = a + b -1 for a, b E Z, show that {Z,*} is an…

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Q: QI/AI Prove that the mathematical system (s), where (o) is an usual composition map is a group? B/…

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