36) Prove that if G/Z(G) is cyclic then G is abelian. [If G /Z(G) is cyclic with generator xZ(G), show that every element of G can be written in the form xaz for some integer a E Z and some element z E Z(G).]

Q: Let (G1, ) and (G2, *) be two groups and p: G1- G2 be an isomorphism. Then * G2 is abelian if G1 is…

A:

Q: Let m, n ɛ Z+ such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: Given that m,n∈ℤ+ such that gcdm,n=1. Suppose that there are M and N abelian groups, up to…

Q: Let m, n e Zt such that gcd(m, n) = 1. Suppose there are M and N abelian groups, up to isomorphism,…

A: We have M and N are abelian groups of order m and n. so, we know that MN is abelian group.

Q: abelian group is not cyclic if and only if it contains a subgroup isomorphic to Zp×Zp

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Q: Prove that the fundamental group is abelian if and only if each homomorphism γ∗ as above only…

A: Let us assume that π1(X), the fundamental group is abelian. Let us consider a loop α with…

Q: Find two elements of maximum order in the group G = Z100 Z4 O Z2. How many such elements are there?…

A: As Z100 has no element of order 100 otherwise it will be cyclic which is not true. So, we consider…

Q: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

Q: (c) Prove that if G is a (not necessarily abelian) group, a, b e G, and a² = b² = (ab)² = e, then ab…

A: Use property of group and solve it.

Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

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Q: If G is abelian then G/Z(G) is cyclic. *True *False

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Q: Prove that if |G|= pq, where p and q are primes (not necessary distinct), then either G is abelian…

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Q: let G be a grup and let a an element of G. prove that is abelian

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Q: 6. Let G be a group with the following property: "If a, b, and c belong to G and ab=ca, then b = c."…

A: For a group to be abelian it should follow that ab=ba for all a,b belonging to G. Now we will…

Q: Let (G1, ) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then O G2 might be abelian…

A: G2 might not be abelian even if G1 is abelian.

Q: Let (G1, •) and (G2 , *) be two groups and o: G1→ G2 be an isomorphism. Then * O G2 might be abelian…

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Q: is] Let G and H be groups, and let T:G→H_be Isomorphism. Show that if G is abelian then H is also…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: ab 2. Show that (Z, *) is an abelian (commutative) group, where' is defined as a*b = and Z is the…

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Q: Let (G1, •) and (G2 , *) be two groups and :G1 G2 be an isomorphism. Then * G2 might be abelian even…

A: we know that,  If two groups are isomorphic, then all the property will be similar i.e. one  is…

Q: Show that G ⨁ H is Abelian if and only if G and H are Abelian.State the general case

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Q: 18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic if and only if…

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Q: b. Find all abelian groups, up to isomorphism, of order 720.

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Q: Let (G1, •) and (G2, *) be two groups and p: G1→ G2 be an isomorphism. Then * G2 might be abelian…

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

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Q: Exercise 8.1.15. Suppose N G. If N is abelian, and G/N is abelian, does G have to be abelian? Give a…

A: Suppose N⊲ G. If N is abelian, and G/N is abelian, Now we  prove that G is abelian with the help of…

Q: IfGis a finite group and some element of G has order equal to the size of G, we can say that G is:…

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Q: Suppose that G is an Abelian group with an odd number of elements.Show that the product of all of…

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Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb¬ va,bEG

A: We need to prove that a group G is abelian if and only if (ab)-1=a-1b-1 , for all a,b in G. The…

Q: If a, b are elements in an abelian group G, show that (ab)n = anbn  for every positive integer n.

A: It is given that 'a' and 'b' are elements in abelian group G. This means (abn)=(anb).   Let for n=1

Q: Show that every abelian group of order 255 (3)(5)(17) is isomorphic to Z55 and hence cyclic. [Ilint:…

A: We have to solve given problem:

Q: 6. Let H and K be subgroups of G. Suppose |H| = 35 and |K| = 28. Prove %3D that HNK is abelian.

A: The intersection of two subgroups is a subgroup. Also, recall that the prime order subgroup is…

Q: Prove that a group G is abelian if and only if (ab)-1 = a¬b¬1 va,bEG

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Q: 8. Prove that Q is not a subcartesian product of infinite cyclic groups.

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Q: Let (G1, •) and (G2 , *) be two groups and p : G1 – G2 be an isomorphism. Then * O G2 is finite if…

A: We will use basic results of group theory

Q: 3.10. Let G be a group containing at most three elements. Show that G is abelian.

A: Here given that G is a group contains at most three that is G contains 1 or 2 or 3 elements. Case -…

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

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Q: Prove that a group G is abelian if and only if (ab)-1 = a-lb-l Va,bEG Attach Filo

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Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

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

A: A group G is abelian if it is commutative under the operation *. In other words, G,* is an abelian…

Q: (а) (Q, +) (b) (Zs, ·)

A: (a)(ℚ,+)(b)(ℤ8,.)

Q: Prove that every group of order (5)(7)(47) is abelian and cyclic.

A: See the attachment

Q: 1) Prove that if G is an abelian simple group then G = Z„ for some prime p. (Do not assume G is…

A: By assuming G is simple abelian group we have to prove that G is isomorphic to Zp.

Q: A group G in which (ab)? = a²b² for all a, b in G, is necessarily O Abelian Of order 2 O Finite

A: Since it has associative property (ab) ^2  = a^2b^2

Q: Prove that a group G is Abelian if and only if (ab)-1 = a-1b-1 forall a and b in G.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: Suppose G is a group in which all nonidentity elements have order 2. Prove that G is abelian.

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Q: Prove that if G/Z(G) is cyclic, then G is abelian.

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Q: Let (G1, •) and (G2 , *) be two groups and p : G1→ G2 be an isomorphism. Then * G2 might not be…

A: G2 is abelian if G1 is abelian.

Q: Let m and n be integers that are greater than 1. (a) If m and n are relatively prime, prove that Zm…

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