Let H₁, H₂ and H, be abelian groups. Prove or disprove: H₁ xH₂x H, is an abelian group.

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: (Z, +) is a group and infinite group

A: Let a binary operation '*' defined on a set G, then it forms a group (G,*) if it holds the following…

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

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Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

Q: If ?^2 = ? for each element “a” of a group G, where “e” is the identity, then prove that G is…

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Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: The Galois group of  x4 +x2 +1 is the same as that  of  x6 -1 and is of order 2.

A: Since the primitive nth root of unity is        e2πi/n=cos2πn+i sin2πn. Then primitive third root…

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

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Q: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

Q: (True/False) Suppose :ZZ4, x→ 2x (mod 4). Then ker() = 42. (True/False) Suppose G is an abelian…

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Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

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Q: (a) (Z,*) with a × b = a/b is (b) (Q, *) with a * b = ab - a - b is (c) (Q,*) with a*b = a² +6² is…

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Q: = n and o(b) = m Let G be a group and suppose a, b E G satisfy ab = ba. Assume that o(a) : with (n,…

A: Given G is a group, such that a, b∈G satisfy ab=ba. Also, we have oa=n and ob=m with n, m=1.   To…

Q: set P(S)

A: According to the question, The power set P(S) of a set S is the set of all subsets so, of S,…

Q: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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

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Q: not

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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: Construct the Cayley table for (Zo) ,c), and verify that this is an Abelian group.

A: 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3…