Let (G1, •) and (G2 , *) be two groups and o: G1→ G2 be an isomorphism. Then * O G2 might be abelian even if G1 is abelian G2 might not be abelian even if G1 is abelian. G2 is abelian if and only if G1 is cyclic. O G2 is finite if G1 is finite.

Q: 21. Suppose a is an element of order rs in an abelian group G. Prove that if r and s are relatively…

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

Q: Any subgroup of F2 (the free group upon 2 generators) is not abelian." Is this statement true or…

A: The given statement is Any subgroup of F2 (the free group upon 2 generators) is not abelian.

Q: Suppose that G is an Abelian group of order 35 and every element of G satisfies the equation x35 =…

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Q: Give a careful proof using the definition of isomorphism that if G and G` are both groups with G…

A: Definition :Let, G and G' be any two group . Then a map φ: G →G' is said to be a homomorphism if φ(a…

Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6. Recall that ℤ? denotes the abelian cyclic group of order…

A: It has to be proved that Z12 is not isomorphic to Z2 ⊕ Z6. Isomorphism is defined as one-to-one…

Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

A: We have to show that the permutation of m letters, m≥3 that is Sm is not abelian. For proving the…

Q: 4. Let p be a prime. Find all abelian groups of order på that contain no elements of order p up to…

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Q: Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system…

A: The given set is Zodd=. . . . . . -5, -3, -1, 1, 3, 5, . . . . . . . We have to determine  (i) Zodd,…

Q: Let G= U(32) = {1,3,5,7,19, 1,13,15,17, 19, 21 - H = U(32)= {1,17} %3D %3D Which of the 3 Abelian…

A: We have to find  isomorphic group to G/H:

Q: 25. Let G be the set of all matrices of the form (i), with a € Z and matrix multiplication as the…

A:

Q: 11. Prove that det(AB) = det(A) det(B) in GL₂(R). Use this result to show that the binary operation…

A:

Q: LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

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Q: 6. ! Write down the relation matrix of the abelian group G specified as follows. G - {x, y, z, w |…

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Q: . Find all of the abelian groups a) of order 200 up to isomorphism b) of order 720 up to…

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Q: Give an example of two non-homomorphic finite abelian groups that have the same number of elements…

A: We need to find an example of two non holomorphic finite abelian groups which have the same number…

Q: U16  is the set of elements in Z16 which has inverse in Z16. Prove that U16 is a group under…

A: U16=x∈z16:x-1∈z16An element x∈zn has an inverse in znif and only if…

Q: 6. Let m > 3. Prove that the symmetric group on m letters is not abelian.

A:

Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.

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Q: 1. Show that the group (a, b, c|bª = a, c¬1 = b) is abelian. %3D

A: The objective is to show that the given group is abelian. Given group is: a,b,c|b4=a,c-1=b

Q: Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

A: First, consider that the group is abelian. So here first compute (ab)-1 for a and b belongs to G,…