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

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

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Q: Let m and n be relatively prime positive integers. Prove that the order of the element (1, 1) of the…

A: To prove ℤm×ℤn≅ℤmn ⇔ gcd(m,n)=1 Part 1) Proof of 'only if'  part  ℤm×ℤn≅ℤmn ⇒ℤm×ℤn is cyclic. Let…

Q: Prove that any cyclic group is abelian.

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Q: Prove that every group of order 1225 has a normal abelian Sylow 5-subgroup.

A: Since not a particular question asked as per guidelines solution to only first question  is given…

Q: Let G be a group such that a^2 = e for each a e G. Then G is * О Сyclic O None of these O…

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

A: We know that a finite group G is said to be cyclic if and only if  there exist an element in G such…

Q: Every quotient group of a non-abelian group is non-abelian.

A: (e) False (f) True (g) True Hello. Since your question has multiple sub-parts, we will solve first…

Q: Prove that a group of order 12 must have an element of order 2.

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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: Prove that a group of order 3 must be cyclic.

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: 4. List all of the abelian groups of order 24 (up to isomorphism).

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Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: let G be a group of order p^2 where p is prime. Show that every subgroup of G is either cyclic or…

A: Given that G is a group of order p2, where p is prime.To prove that every subgroup of G is either…

Q: QUESTION 6 Prove that a cyclic group G= is Abelian.

A: Given cyclic group G = <a>

Q: Let G={-1,0,1}. Verify whether G forms an Abelian group under addition.

A: G  is a group under '+' if  (i) a , b E G  -----> a + b E G     (ii) a E G called the identity…

Q: There are two group of order 4, namely Z, and Z2 Z2, and only one of them can be isomorphic to G/Z.…

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

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Q: The center of an abelian group G is: a) {e) b) G c) A cyclic subgroup d) None of these

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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: Let ?1 , ?2 ??? ?3 be abelian groups. Prove that ?1 × ?2 × ?3 is an abelian group.

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Q: Let G be an abelian group of order 2n, where n is odd. Use Lagrange's Theorem to prove that G…

A: Given :               The group G, which is an abelian group of order 2n, where n is odd…

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

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

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

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Q: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

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Q: Prove that (ab)2 = a²b² for all a, b in a group G if and only if G is Abelian.

A: Let G be set and "·" be a binary operation then G,· is a group if  Clouser property. That is, if…

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: 1. Prove or disprove (h) There is non-abelian group of order 255. i) There is a simple group of…

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Q: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.

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Q: Q 12 Every subgroup of an abelian group is abelian.

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

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Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: Let G be a group. V a, b, c d and x in G, if axb cxd then ab = cd then G is necessa: Abelian Of…

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Q: Prove that is (ab)-1 = a-1b-1 for all a,b in group G, then G is abelian.

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Q: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

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Q: Let M be a group (not necesarily an Abelian group) of order 387. Prove that M must have an element…

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Q: Let G be a group such that a^2 = e for each aEG. Then G is * Non-abelian Cylic Finite Abelian

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Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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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: Let G be a group and suppose that (ab)2 = a²b² for all a and b in G. Prove that G is an 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 G1 and G2 be groups. Prove that G1 × G2 is abelian if and only if G1 and G2 are both abelian.

A: Abelian groups :  A group 'G' is abelian if ,                  ab = ba   ∀ a,b ∈ G  Direct product:…

Q: Let G, H, and K be finitely generated abelian groups. Prove or disprove: If G × H ∼= K × G, then H…

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