Q: THE SET (1, 2, 4, 7, 8, 11, 13, 14) IS A GROUP UNDER MULTIPLICATION MODULO 15. THE INVERSES OF 4 AND…
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Q: for each of the group Z2,Z4,Z6 INDICATE WHICH CYCLIC. FOR those cyclic list all the generators?
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Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…
A: Suppose G is a cyclic group with an element with infinite order. It means that order of group is…
Q: Give all the possible elementary divisors of a group of order 40.
A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5
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: 6. Give an example of two groups with 9 elements each which are not isomorphic to each other (and…
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Q: 4. If a is an element of order m in a group G and ak = e, prove that m divides k. %3D
A: Step:-1 Given that a is an element of order m in a group G and ak=e. As given o(a)=m then m is the…
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: 5. Find the number of generators of the cyclic group Z15
A: To find the number of generators of the cyclic group ℤ15.
Q: 11. Prove that every Cayley table is a Latin square for a group. That is, each element of the group…
A: To prove, each element of the group appears exactly once in each row and each column of a Cayley…
Q: 1. Show that the set {5, 15, 25, 35] is a group under modulo 40. What is the identity element of…
A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…
Q: 1. What are the generators of the group Z60?
A: We have to find the generator of the group ℤ60
Q: How many elements of a cyclic group with order 10 have order dividing 5?
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Q: Show that if n > 6, then the symmetric group Sn contains an element of order greater than n
A: According to the problem, we have
Q: How many proper subgroups are there in a cyclic group of order 12?
A: let G be a group of order 12 and let x be the generator of the group. Then the group generated by x,…
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: many
A: We have to find the number of generator of the given group of order
Q: 1.Show that the set {5,10,25,35} is a group under multiplication modulo 40 by constructing its…
A: Let us denote the operation given in the question, multiplication modulo 40, with · and the usual…
Q: Analyze the properties of Zs with multiplication modulo 6 to determine whether or not this operation…
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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 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: How many elements of the cyclic group GF(81)* are generators?
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Q: 16. Determine whether the set {1, 2, 3, 4} with the opera- tion multiplication modulo 5 forms a…
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Q: What is the order of the element $(\overline{2}, \overline{9})$ in $Z_{4} \times U_{10}$ is (…
A: The set ℤ4 ,i.e the set of congruence modulo 4 is an additive group. Add the element 2 with itself…
Q: 1. Determine all subgroups of the group (U13, ·)
A: The sub group of U13 is to be determined.
Q: The cyclic group of order 12 acts on {1,2,..., 12} with the following cycle structure. (1)…
A: Given that, the group of order 12 In this case of necklace, there is no difference between…
Q: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?
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Q: The set {1, 2, 4, 7, 8,11,13,14} is a group under multiplication modulo 15. T inverses of 4 and 7…
A: Introductions :
Q: Every commutative group has at least element ??
A: Every commutative group has at least element ? We know that , every commutative group…
Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140
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Q: The alternating group A5 has 5 conjugacy classes, of sizes 1, 12, 12, 15, 20. Use this information…
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Q: 50. How many proper subgroups are there in a cyclic group of order 12? A 4 в з с 2
A: see 2nd step
Q: List all elements of the group U(15). Is this group cyclic?
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Q: 3. List all elements of the cyclic subgroup of Z12 generated by 5
A: Solving
Q: order 8 of a group G, and =
A: Given that order of a is 8 .Then a8=e Rearrange a little bit , we can have a42=e Hence order of…
Q: Show that the set {5, 15, 25, 35] is a group under modulo 40. What is the identity element of this…
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Q: Let p be a prime number and (G, *) a finite group IGI= p?. How can you prove that the group (G, *)…
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Q: Suppose that G is a cyclic group and that 6 divides |G|. How manyelements of order 6 does G have? If…
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Q: c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…
A: #Dear user there is a mistake in the question the assumption is for the element c and d of a group…
Q: {a3 }, {a2 }, {a5 }, {a4 } Which among is not a subgroup of a cyclic group of order 12?
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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: A cyclic group is abelian
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Q: Find the order of the group and the order of each element in the group. In each case, how are the…
A: The no. of elements present in a group is the order of the group. n is the least positive integer…
Q: Suppose G is a cyclic group with an element with infinite order. How many, elements of G have finite…
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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=<g> for some g in G, where…
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: Find an example of a noncyclic group, all of whose proper subgroupsare cyclic.
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Q: True or false? Every group of 125 elements has at least 5 elements that commute with every element…
A: Let G be a group whose order is 125 ⇒G=125=53 Center of a group G ( ZG ) is the set of all those…
Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…
A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…
Q: Give the Cayley table for the group Z2 under multiplication modulo 12.
A: Since , We know that Z12 = 0,1,2,3,4,5,6,7,8,9,10,11 and…
Q: 6.7 Construct a nonabelian group of order 16, and one of order 24.
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Q: construct a cayley table for the dihedral group of order 5
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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: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…
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