Let a be an element of order 8 in a group G. 6 Then order of a Select one: O a. 6 о Б.48 C. 2 d. 4

Q: Q1. Decide which of the following sets with given binary operations are groups? (i). Q* = { € QI >…

A: NOTE: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question…

Q: In Exercises 11 and 12, part of the multiplication table for the group G = {a, b, c, d} is given. In…

A: To find- Part of the multiplication table for the group G = a, b, c, d is given. In each case,…

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: Determine whether the following is a semigroup, monoid, or a group. 1. G=Z, a*b = a + ab 2. G=2Z,…

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Q: Find the order of each of the following elements in the multiplicative group of units Up. a. [2] for…

A: Consider the multiplicative group U13. Find the order of element 2.   The identity of a…

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: If a is an element of order 8 of a group G, and = ,then one of the following is a possible value of…

A: Given that a is an element of order 8 and a4=ak

Q: O In a group (G,x), if IGl= 37, the number of Passible Subgroups in G are

A: Note: Since you haven't mentioned which question you would like to get answered. We are providing…

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: Let a be an element in group G, order (a)=15, then the order a10 = 3, where a10 E G

A: Use the formula of o(a^r) , if o(a) is given.

Q: 1. Determine all subgroups of the group (U13, ·)

A: The sub group of U13 is to be determined.

Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * e b с a e e а…

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Q: 10) Which of the following is a group? * O (Z,*), a* b = a + b - 1 va, b e Z O (Z,*), a* b = a - 2b…

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Q: Which of the following is a group? O O

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Q: The elements of order 8 in the group (Zg ,+) is a)4 b) 5 c)6 d) 8

A: In this question we have to find the number of elements of order 8. (a) 4 is correct option

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 subgroups of the group G = (Z30, +30)? O a. 4 O b. 7 O c. 5 O d. 6 O e. 8

A: We know that τ(n) is number of positive divisors of n Also τpaqbrc = (a+1)(b+1)(c+1) , where p , q ,…

Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * e a b с ее а…

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Q: is a group with identity (eg, eH).

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Q: How many subgroups of the group G = (Z30, +30)? а. 8 O b. 6 С. 5 O d. 7 O e. 4

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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: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

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

A: Concept:

Q: (b) Suppose G is a group, H, K < G, |H|= 30, |K| = 20, and |HN K| = 10. What is |HK|? %3D %3D

A: By theorem, Order of product of two subgroup of finite order       Let the two subgroups be H and K…

Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: 18. Which of the following groups are simple? (a). (Z, +). (b). (Zp, +), where p is a prime number.…

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Q: Construct the group table for U (24). Find the order of each element in U(24).

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Q: Let a,b be two elements in a group. Suppose a, b are of the same er. Let m be the smallest positive…

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Q: If H and K are subgroups of G, |H|= 16 and IK|=28 then a possible value of |HNK| is * O 16 4 8.

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Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…

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Q: 2. What is the order of the element 32 in the group Z36?

A: Modular groups  are cyclic groups. A group G is cyclic if G=&lt;g&gt; for some g in G, where…

Q: What is the identity element In the group G = {2, 4, 6, 8) under multiplication modulo 10? Select…

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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: Let a be an element in group G, order (a)=15, then the order a10 = 3, where a10 e G

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Q: The group ((123)) is normal in the symmetry group S3 and alternating group A4.

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Q: Exercise 5 Is every group f ordler u 6) n< 60, (c) n=300 solvable (a) n=1000,

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Q: If H and K are subgroups of G, |H|= 16 and IK|=28 then a possible value of |HNK| is * O 16 6. 4 O O…

A: H and K are subgroups of G H=16 and K=28 we have to find the possible value of H∩K

Q: Given the group (S = {a,b,c, d},8) %3D a d a b a b d - d The inverse of an element z, z e Sis a a с…

A: The solution is given as

Q: а) The set generated by The elemenb a, is agroup G- Suppose G is a group and a be an element of G…

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Q: Compute the orders of the following groups.a. U(3), U(4), U(12)b. U(5), U(7), U(35)c. U(4), U(5),…

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Q: (b) How many elements of the permutation group Se map 2 to 2 and 5 to 5, while the re- maining…

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Q: (b) Consider the incomplete character table for a group given below:

A: The table for the incomplete character is,   1 a b c d x1 1 1 1 1 1 x2 1 1 -1 -1 1 x3 1 1…

Q: Consider a normal subqroup H of order 4 of the dihedral group b2-lab)? => ther Dus< a, b: Q= n prove…

A: Write the general formula for the element of the group and use operation

Q: (b) Complete the following character table of a group of order 12: 1 3 4 X1 X2 X3 4,

A: The character table of a group of order 12:

Q: Consider the multiplicative group ℤ∗8663. a) How many primitive elements does this group have? b)…

A: (a): we have 8540 primitive elements in this group (b): probability is:0.98

Q: Let a belong to a group and |a| = 100. Find |a98| and |a70|.

A: Given: a=100 we know that am=am

Q: 5. Let a be an element of order n in a group and let k be a positive integer. Then =< a™dlnA)

A: To prove : ak=agcd(n,k) Let set d = gcd(n,k)  and then write k=dr by definition of gcd, We prove…

Q: If G is a group and a, b, c € G then (abc) 1 = a'b¯lc=1

A: Use Reversal law of inverse in group theory.

Q: (a) Determine a complete list of elements of order 12 in the group Z240.   (b) What is the order…

A: Z240 is a cyclic group.Number of element of order 12 is ϕ(12) .…