The group (Z6,6) contains only 4 subgroups

Q: Construct the group table for (Z,, + ,).

A: We have to construct the group table for (Z5,+5). Where Z5 is modulo 5 group of integers. So to…

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: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

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

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Q: Give the lattice of subgroups of (Z2,), of (S,,0), and of (Z(20) ,)

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Q: If a1, a2,...,an belongs to group .what is inverse of a1a2...an

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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: 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: Explain why S8 contains subgroups isomorphic to Z15, U(16), and D8.

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

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Q: How many elements of a cyclic group with order 10 have order dividing 5?

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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: 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: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: Draw the subgroup lattice for Z28-

A: Draw the subgroup lattice for Z28

Q: Write the group table for Z₂ XS2.

A:

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: How many subgroups of a not abelian group of order 6 is non-cyclic? Select one:

A: Given: The order of the group = 6.

Q: Z, x is not a group.

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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: Find all the subgroups of Z48. Then draw its lattice of subgroups diagram.

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Q: Determine all cyclic groups that have exactly two generators.

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Q: What are the three things we need to show to prove that an ordered pair is a group?

A: We have to give the properties of an ordered pair to prove that it is a group.

Q: Is it possible to find a group operation e on a set with 0 elements? With 1 element? Explain why or…

A: The question is :: is there possible to find a group operation on a set of 0  element? Or with 1…

Q: How many elements does the group of rotations of a regular hexagon have?

A: The group of rotations of a regular hexagon is D6. And as the order of the group Dn is 2n.

Q: If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?

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Q: Find the three Sylow 2-subgroups of S4

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Q: Give three examples of groups of order 120, no two of which areisomophic. Explain why they are not…

A: Let the first example of groups of order 120 is, Now this group is an abelian group or cyclic group…

Q: Construct a subgroup lattice for the group Z/48Z.

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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: Show that a finite group of even order that has a cyclic Sylow 2-subgroup is not simple

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Q: 3. How many cyclic subgroups does S3 have?

A: The objective is to find the number of cyclic subgroups of S3. Subgroups of S3 are, H1=IH2=I, 1…

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: Write U(32) as the internal direct product of two proper subgroups.

A: Given:         U32 We use the theorem namely Ut≈Usst   because Usst is a subgroup of Ust

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Draw the lattice of the subgroups Z/20Z.

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Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

Q: How many subgroups does Z/60Z have?

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Q: What limits will include the middle 80% of the group?

A: The required calculation can be defined by the formula Z=x-μσP(a≤Z≤b)=P(b≤Z)-P(a≤Z) Given, What…

Q: Find two p-groups of order 4 that are not isomorphic.

A: Consider the groups ℤ4  and  ℤ2⊕ℤ2. Clearly, both of the above groups are p-groups of order 4.

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: Find the center and the commutator subgroup of S3 x Z12-

A: Solution

Q: The inverse of 3 in the group (Z5, o5) is

A: Definition:

Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

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Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

Q: b. Find all the cyclic subgroups of the group ( Z6, +6).

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Q: a) How many subgroups does (Z10,O) have? What are they? b) How many subgroups does (Z74,Ð) have?…

A: 2. a) Consider the group ℤ10, ⊕. The elements of the above group are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9…