Consider the multiplicative group ℤ∗8663. a) How many primitive elements does this group have? b) What is the probability that a randomly chosen member of this group is a primitive element?

Q: In the additive group of the integers modulo 24, the order of 8 is a) 16 b) 32 c) 3 d) 1 e) 8

A: We need to find the order of 8 in the additive group of the integers modulo24.

Q: Sylow-5 subgroups-

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Q: If n is not prime, then G = {1, 2, 3,..., n-1} is not a group under multiplication mod n.

A: We prove this result by contradiction.  Let n is not prime and G={1,2,3,....,n-1} is a group under…

Q: You are given a prime p = 4969 and the corresponding multiplicative group Z4969*. If possible, a…

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Q: list the elements of the subgroup generated by the subset {18, 24, 39} of Z

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Q: If p is a prime number the set {1,2,3,...,(p-1)}  is a group under multiplication modulo p.

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Q: any permutation o and its inverse arity.

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Q: Create the table and the subgroup diagram of the following:   a. Z4   b. V-Klein 4-group

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Q: What is the order of the symmetry group in R3 of a 4-prism (a boxwith square ends that is not a…

A: Given: Symmetry group in R3 of a 4-prism.To find: The order of given symmetry group.

Q: 6. Compute the m-adic representation of 526 for m = 2,3,7, and 9.

A: We have to compute the m-adic representation of 526 for m = 2, 3, 7 and 9.

Q: 5. Describe the symmetries of a square and prove that the set of sym group. Give a Cayley table for…

A: The symmetries of a square refer to the various transformations and rotations that can be applied to…

Q: Which of the following mathematical systems are semigroups? Which are groups? (N, ∗), where a ∗ b =…

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Q: Let Sn be the group consisting of the set of all permutations on n letters, together with the…

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Q: In a group G,let a,b and ab have order 2.show that ab=ba

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Q: Is a permutation of one element considered to be an even permutation? For example, is (1) an even…

A: To clarify the notion of an even permutation

Q: Show that the set (5,10,25, 35} is a group under multiplication modulo 40 by constructing its Cayley…

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Q: Let G be a finite nonabelian group. Pick (uniformly at random) an element (a,b,c) ∈ G³. What is the…

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Q: frove that the intersection et two subgroups of a group G is atso a subgroup of G

A: Let G be the group and H and K be the subgroups of Gclaim-: HnK is sub group since e∈H…

Q: How many elements of order 33 are there in the dihedral group 33 ?D

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Suppose chips are randomly drawn from a bag, and each chip is labeled with either 1, 2, 3, 4, 5, 6,…

A: S={1, 2, 3, 4, 5, 6, 7, 8, 9} n(S) = 9 E = odd number = {1,3,5,7,9} n(E) = 5

Q: A square has four rotational symmetries (including the identity) and four line symmetries. Make a…

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Q: Determine the character table of the klein four group.

A: Klein four group is a group containing four elements, That is, for, G=e, a, b, c, where e is the…

Q: ) The lower class limit for the group between 11 and 19 is ____________?   a) 10.5 b) 11 c)…

A: Ans- The lower class limit for the group between 11 and 19 is ____________

Q: The set G = {1, 5, 7, 11, 25, 29, 31, 35} is a group under multiplication modulo 48, with the group…

A: Here is an additional explanation(a) How many elements of order 2 does \( G \) have?An element \( g…

Q: Every commutative group has at least element ??

A: Every commutative group has at least               element ?  We know that , every commutative group…

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: explain in detail if the following statement is true or false: " In the permutation group S5, the…

A: First we define the permutation as follows:Let  be a finite set having  elements. Let  be a…

Q: Present the formulation of the following concept: Partition of n objects into r groups with i-th…

A: A set containing n objects is partitioned into r non-empty subsets or groups and each group i has ni…

Q: Which groups Un are cyclic for n in the range 50 ≤ n ≤ 60?

A: For the range 50≤n≤60, the cyclic groups Un​​ are those with n=53and n=59, since these are prime…

Q: Consider the multiplicative group Z4547. a) How many primitive elements does this group have? b)…

A: Given thatMultiplicative group We have to find the number of primitive elements in the given…

Q: Up to isomorphism, how many distinct groups G are there with order 1? 2? 3? 4? (Create possible…

A: Group of order 1 is, up to, upto isomorphism we get only one group of order 1 .Group of of order 2…

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

A: We have to find Sylow 5-subgroups of S_5

Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple. .

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

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…