In the group Z, find а. (8, 14); b. (8, 13); с. (6, 15); d. (m, п); е. (12, 18, 45). In each part, find an integer k such that the subgroup is (k).

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

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

Q: compute the 3 -sylow subgroups of S5

A: n3 (S5 ) =(1+3k) /40

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

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

Q: Q/ How many non-trivial subgroups in s, ? a) 2 b) 3 c) 4

A: S3 is the set of permutations on the set {1, 2, 3}S3={e, (1 2), (1 3), (2 3), (1 2 3), (1 3 2) }The…

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

A:

Q: Find cyclic subgroups of S4 that have three different orders.

A: There are more subgroups than just the cyclic ones.   Trivial:  there is <e> = {e}.   There…

Q: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

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

A:

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

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Q: Find a subgroup of Z12 ⨁ Z18 that is isomorphic to Z9 ⨁ Z4.

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 K is a proper subgroup of H and H is proper subgroup of G. If |K| = 42 and |G| = 420,…

A:

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

A:

Q: (e) Find the subgroups of Z24-

A: Given that

Q: O Find Subgroup of order a, 3,4 and 6

A:

Q: Show that Z has infinitely many subgroups isomorphic to Z.

A: We have to show that Z has an infinitely many subgroups isomorphic to Z.

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: Find the three Sylow 2-subgroups of S4

A:

Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

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Q: Find all of the distinct subgroups of Z90 and draw its subgroup diagram.

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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: Find the number of ways to colour faces of a cube using at most k colors, up to the group of…

A: First you enumerate the elements of the symmetry group of the cube. There are 24 of these, falling…

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

A:

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: Find all generators of the subgroup of Z/60Z with order 12.

A: Generators of the group

Q: Show that every subgroup H of the group G of index two is normal.

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

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Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

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Q: Find a subgroup of order 4 in U(1000).

A:

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

A:

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: 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 all cosets of the subgroup of Z12.

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

A:

Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

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: Determine if B is a subgroup of A

A: Introduction: Subgroup is a part of a group. More precisely, a subgroup is a non-empty subset of a…

Q: Find the subgroup of ℤ generated by the subset {6,9}.

A:

Q: 1- (Z,+) is a subgroup of (Q,+) 2- (Q,+) is a a subgroup of (R, +) 3- (R,+) is a a subgroup of (C,+)…

A:

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

A:

Q: b. Find the center and the commutator subgroup of S2 x Z7.

A: Now we knew that  Z2 is isomorphic to S2. So it is commutative group. The center subgroup of G := S2…

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…