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

Q: compute the 3 -sylow subgroups of S5

A: n3 (S5 ) =(1+3k) /40

Q: 8. Find a non-trivial normal subgroup of the octic group. Demonstrate that this subgroup is normal.

A: According to the given information, it is required to find a non-trivial normal subgroup of the…

Q: How many non-trivial subgroups in S3? 3 4 5

A: Trivial subgroup: A subgroup containing identity element is called as trivial subgroup and other…

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

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Q: When do you use a t-test for independent groups and what do the numerator and deniminator in the…

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Q: How many subgroups of are equal to the subgroup ?

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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: Define :normal subgroup

A: To write the definition of normal subgroup

Q: Prove that {(1), (1, 2)(3, 4), (1, 3)(2,4), (1, 4)(2,3)} is a subgroup of S4.

A: We use the Cayley table.

Q: Define Group theory ?

A: To define group theory

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

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Q: How many subgroups of order 4 does D4 have?

A: Solution: Assume the following dihedral group D4, The above group consists of three subgroups of…

Q: Show that the set of rational numbers (positive and negative fractions) together with multiplication…

A: The set under question can de defined as S={pq|p∈ℤ,q∈ℤ,q≠0}We have to show that Set S along with…

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

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Q: How many non-trivial subgroups in S3? 3 4 2.

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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: 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: Give an example of a group that has exactly 6 subgroups (includingthe trivial subgroup and the group…

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Q: How many non-trivial subgroups in S3? O 2 O 4 5 3.

A: To find the number of non trivial subgroups of S3.

Q: In practice, what can be the differences between the relationships of two pairs of groups, knowing…

A: Homomorphism -  It is  algebraic term for a function having some algebraic operations. For a group…

Q: he first effective use of groups was in the early nineteenth century by

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

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Q: How many subgroups of a

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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: How many Sylow 5-subgroups of S5 are there? Exhibit two.

A: We have that |Sn| = n! Hence, |S5| = 5! = 120 = 23.32.5 By Third Sylow Theorem the number of Sylow-5…

Q: How many non-trivial subgroups in S3? 4 O 5

A: Given group is S3 S3=3!=6 In total, there are 6 elements in the group S3 Elements of the given group…

Q: How many cyclic subgroups does have U(15) have? 4 3

A: We will determine the cyclic subgroup generated by each element of G

Q: Find the number of G-orbits in X := {1,2,3, 4, 5, 6, 7, 8, 9, 10}, where G < S10 is the subgroup…

A: Solution:Let g1 = (1,2), g2 = (3,4,5) - and g3 = (6,7,8,9). All three cycles at sight are…

Q: How many subgroups are there in Z/14Z?

A: Subgroup: Let (G, ∙) be a group and S be a non‐empty subset of G.Then, S is called a subgroup of G…

Q: Find four different subgroups of S4 that are isomorphic to S3.

A: If there exists an isomorphism between the subgroups they are called isomorphic ( Isomorphism is…

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: 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: 2- Find the cyclic subgroups in (Z,, +,) generated by 3,5.

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Q: If G is an infinite group, what can you say about the number ofelements of order 8 in the group?…

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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: 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: 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: Define group in abstract algebra

A: As We know,

Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

Q: a subgroup of (G, *) , known as the normalizer subgroup of (H, *) in (G, *).

A:

Q: How many Sylow 3-subgroups of S5 are there? Exhibit five

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Q: Create Cayley table for square isometry groups

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Q: Construct the distinct left cosets of the subgroup {(1), (1,3,2), (1,2,3)} in S3.

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Q: In Z24 the number of all subgroups is 8 O 6.

A: We are asked to find the subgroup of z24

Q: How many subgroups are there in D4? Also find them.

A: D4 has 8 elements:. {1, r, r2, r3, d1, d2, b1, b2,}. where r is the rotation on 90◦, d1, d2 are…

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…