let G={e, a, a^2, a^3, a^4, a^5, a^6, b, ab, a^2b, a^3b, a^4b, a^5b, a^6b}. related by the equation ba=a^2b, fined order of each element and proper nontrivial subgroups of G.

Q: In (Z12, +12) , H, = {0,3,6,9} and H2 = {0,3} are tow subgroups of the group (Z12, +12), but (H, U…

A: We have given   H1=0,3,6,9 and H2=0,3 are two subgroups of the group Z12, +12. We can see here…

Q: Compute the factor group (Z6 x Z4)/(([2]6, [2]4))

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Q: Write ?(16) as an internal direct product of its subgroups.

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Q: List two examples of nontrivial proper subgroups of the indicated group. a) Z18 b)U(18)

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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: 7. Show that 4 is a subgroup of S,

A:

Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: Express U(165) as an internal direct product of proper subgroups infour different ways

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Q: Show that the intersection of two normal subgroups of G is a normalsubgroup of G. Generalize

A: Given: To prove the intersection of two normal subgroups of G is a normal subgroup. Consider G be a…

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: List all subgroups of Z15

A: We have to list all subgroups of Z15. Note that, Z15 is a cyclic group of order 15 and the set of…

Q: Prove or give counter example Every characteristic subgroup is fully invariant

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Q: Explain why the only simple, cyclic groups are those of prime order.

A: Proof: Let G be a simple group with |G|>1. We want to prove that G is a cyclic group of prime…

Q: How many nonisomorphic abelian groups of order 80000 are there?

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Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

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Q: If G = {1, 2, 3, 4, 5, 6} is the upper group of multiplication modulo 7, H is a subgroup of G with H…

A: Given: Group G=1,2,3,4,5,6 under multiplication modulo 7. Subgroup H=1,2,4 of G. To find:   a) Right…

Q: The elements of the quotient group (2/8).) are {} 2- -208,-108,008,108,208,...) 3- (0,1,2,3,...,8)…

A: Given that the quotient group Z8.⊗. We have to find the elements of the quotient group Z8.⊗.…

Q: Determine all subgroups of R* (nonzero reals under multiplication)of index 2.

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Q: Suppose that H is a proper subgroup of Z under addition and that Hcontains 12, 30, and 54. What are…

A: It is given that H is a proper subgroup of Z under addition and that H contains 12, 30 and 54.

Q: Find all inclusion between subgroups in Z/48Z

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Q: The intersection of two normal subgroups is always normal sub group. (True/False)

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Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

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

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Q: Prove that a group of even order must have an odd number of elementsof order 2.

A: Given: The statement, "a group of even order must have an odd number of elementsof order 2."

Q: Give an example of an infinite non-Abelian group that has exactlysix elements of finite order.

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Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: The order of identity element equal itself in any group

A: .

Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 4x. Ø-1(4) None of the choices O {1, 16, 31,…

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Q: {a3 }, {a2 }, {a5 }, {a4 }   Which among is not a subgroup of a cyclic group of order 12?

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Q: Let Ø: Z50 → Z15 be a group homomorphism with Ø(x) = 3x. Then, Ker(Ø) = * (0, 5, 10} None of the…

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Q: Example: Show that (Z,+) is a semi-group with identity

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Q: EQ6/ Give example of a group such that has element of order 7 and nurmal subgruup of Order 3.

A: A detailed solution is given below.

Q: List six examples of non-Abelian groups of order 24.

A: The Oder is 24

Q: Show that the groups Z8xZ20xZ12 and Z120xZ4xZ4 are isomorphic by define a one-one and onto map? what…

A: We will use the basic knowledge of groups and abstract algebra to answer this question.

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: The group (Z6,6) contains only 4 subgroups

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Q: What are the eight steps in computing a t-test for dependent groups

A: The paired t is used when the two sample are paired or related samples. When the sample is measured…

Q: Prove that the intersection of two subgroups is always a subgroup.

A: In this question, we prove the intersection of the two subgroup of G is also the subgroup of G.

Q: Let Ø:Z50 Z15 be a group homomorphism with Ø(x)=4x. Then, Ker(Ø)= O (0,15,30,45} O None of the…

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Q: The group (Z6,+6) contains only 4 subgroups

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Q: Check whether a group of order 156 is simple or not.

A: Definition: A group G is said to be simple if the only normal subgroups of G are the trivial group e…

Q: What is the numbers group of

A: The given number is 5. The value of 5 is 2.23606...

Q: n point group D2d, for each of the irreducible representations, verify that the sum of the squares…

A: In point group D2d portions of the D2d character table D2d E 2S4 C2 2C2' 2□d A1 1 1 1 1 1 A2…

Q: Let alpha, beta in S8 ( Symmetric group) where alpha=(1,8,5,7)(2,4) and beta=(1,3,2,5,8,4,7,6).…

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Q: 2) Given example of an infinite group in which every nontrivial subgroup is infinite.

A: Let G=a be an infinite cyclic group generated by a, whose identity element is e. Let g∈G, g≠e,…

Q: Show that a group of order 12 cannot have nine elements of order 2.

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

Q: (5) Make a list of all group homomorphisms from Z4 → Z8.

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Q: Suppose that G is cyclic and G = (a) where Ja| = 20. How many subgroups does G have?

A: