The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

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Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350.  What are the…

A: To determine the possible subgroups H satisfying the given conditions

Q: The following is a Cayley table for a group G. The order of 4 is: 2 3 5 2 3 4 5 3 4 1 2 4 2 1 3 2 3…

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: Find all the conjugate subgroups of S3, which are conjugate to C2 .

A: Given-S3 To find- all the conjugate subgroup of S3 which are conjugate

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

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Q: Are groups Z×10and Z×12 isomorphic

A: Concept:

Q: What is the smallest positive integer n such that there are three nonisomorphicAbelian groups of…

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Q: (c) Prove that the intersection of any three subgroups is a subgroup while the union of two…

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Q: 2. Determine the number of elements of order 15 in the group Z75 ZL20- Also determine the number of…

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Q: Create the table and the subgroup diagram of the following:   a. Z4   b. V-Klein 4-group

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Q: Find a noncyclic subgroup of order 4 in U(40).

A: Let U(40) be a group. Definition of U(n): The set U(n) is set of all positive integer less than n…

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

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Q: Which of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?

A: Since 0 does not divides 12.

Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: Suppose that H is a subgroup of Z under addition and that H contains250 and 350. What are the…

A:

Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350. What are the…

A: Given H be a subgroup of (Z,+) containing250 and 350Note that,  GCD(250 , 350)=1⇒by property of GCD,…

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: Explain why every subgroup of Zn under addition is also a subring of Zn.

A: Every subgroup of Zn under addition is also a subring of Zn as it follows the 1) Associative…

Q: There is a group G and subgroups A and B of orders 4 and 6 respectively such that A N B has two…

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Q: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…

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Q: ] Let N be the subgroup of (Z15, +) generated by 3, that is N = . i- What is the order of N+ 2 in…

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Q: Is the set Z* under addition a group? Explain. Give two reasons why the set of odd integers under…

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Q: Find the three Sylow 2-subgroups of S4

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Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…

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Q: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: Show that the subgroup generated by any two distinct elements of order 2 in S3 is all of S3.

A: Given that, S3 is a symmetric group of permutations. Thus, S3 has 6 elements. By using Lagrange's…

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

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Q: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

Q: What is the smallest positive integer n such that there are two nonisomorphicgroups of order n? Name…

A: Non-isomorphic groups: Groups that have different Sylow-2 groups are non-isomorphic groups.

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right cóset-4 + 2Z contains the…

A: Given the set of all even integers Ififis a subgroup of (Z, +) The right coset is -4 + 2Z

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: 7 is the correct answer.

Q: If H and K are subgroups of G, Show tht H intersecting with K is a subgroup of G.  (Can you see that…

A: Use the 2-step subgroup test to prove H Ո K is a subgroup, which states that,

Q: Consider the discrete group G of order 8 that has the following Cayley diagram e If we have the…

A: The sequence of operations is fcagec. Each element g of G is assigned a vertex: the vertex set…

Q: Which of the following is nontrivial proper sub- group of Z4? {0, 2} O Diophantus of Alexandria {0,…

A:

Q: Q/ How many non-trivial subgroups in s, ?

A: For the given statement

Q: Q2/ In (Z9, +9) find the cyclic subgroup generated by 1,2,5

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Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: True or False: (a) Two finite non-cyclic groups are isomorphic if they have the same order. (b)Let o…

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Q: Since 11 is an element of the group U(100); it generates a cyclic subgroup Given that 11 has order…

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Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

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Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -6 + 2Z contains the…

A: 10 is the element in the right coset.

Q: Give the Cayley table for the group Z2 under multiplication modulo 12.

A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…

Q: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

Q: Which of the following is nontrivial proper sub-group of Z4? {0,2}, {0,3}, {0,1}

A: We have given a group Z4 ,  and we have to find which of the following sets {0,2},{0,3},{0,1} is a…

Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3…

A: Need to find intersection of subgroup