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

Q: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

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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: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: true or false, Let a and b be elements of a group G. If a = a −1 and b = b −1 , then ba is the…

A: Given a and b be elements of a group G. a = a−1 and b = b−1

Q: Chapter 2, exercise 6.4: Prove that in a group, the products ab and ba are conjugate elements.

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Q: Let A = {x E R|–1< x < 1}. Define an operation * on A as follows: a * b = a+b 1+ab where a, b E A.…

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Q: Define ∗ on Z by x ∗ y = x + y + 4. Then: (d) Show that every element of Z is invertible with…

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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: 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: Show that (2*,A) is an abelian group. Make sure that you prove associativity, that you identify the…

A: First to prove that the given group is abelian. Here in the given group G is the set of all subsets…

Q: 2. Let G be a group and H1, H2 <G subgroups. (a) Suppose |H1| = 12 and |H2 = 28, prowe H1 n H2 is…

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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: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group

A: Given:  x '= x ,y' = cx + dy d

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: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

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: 19. Prove that the group (Z/11Z)* is cyclic, and find a generator.

A: We have to prove that the group (Z/11Z)* is cyclic, and find a generator.

Q: 1. Prove or disprove (a). The order of the element (2, 3) € Z6X Z15 is 5. (b). The group Z7X Z17 x…

A: As per the guidelines, I am supposed to do the first three sub parts. Kindly post the other…

Q: 3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show…

A: We will solve this by the help of isomorphism

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: Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the…

A: (i) Closed: Yes, the set of even integers is closed under addition, meaning that when two even…

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: Show that any group of order 4 or less is abelian

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Q: When can you say that a group is abelian.

A: Group: Let us consider a nonempty set G and take a binary operation * then (G,*) is said to be a…

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: Let α = (12)(34) and β = (24). Show that the group generated by αand β is isomorphic to D4.

A: It is given that α=1234 and β=24. It is required to show that the group generated by α and β is…

Q: 1. 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: lternating group An is abelian if and only if n <=3.

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Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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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: Prove that a group of order pq is abelian where p = 9 and q= 5.

A: We have to prove that a group of order pq is abelian where p = 9 and q= 5.

Q: Show that G1XG2 is abelian if and only if both G1 and G2 are abelian.

A: We have to show that is abelian if and only if both are abelian.

Q: Show that the translations of Rn form a group.

A: Given: ℝn. We have to show that: The translations of ℝn form a group. We know that the orthogonal…

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: 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: 2. Finite, abelian (and additive), not isomorphic to Z₂ or Z2 x Z2, and also for every element a,…

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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: 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…