2.1. Construct the group under modulo-6 addition. 2.2. Construct the group under modulo-3 multiplication. 2.3. Let m be a positive integer. If m is not a prime, prove that the set {1, 2, ..., m- 1} is not a group under modulo-m multiplication. 2.4. Construct the prime field GF(11) with modulo-11 addition and multiplication. Find all the primitive elements and determine the orders of other elements.

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: Prove or disprove that H = {nEZ\n is divisible by both 8 and 10} is a subgroup of Z. What happens if…

A: Given: H=n∈Z n is divisible by both 8 and 10We need to show whether H is a subgroup of Z or not and…

Q: 19.3.4. Construct a field E with nine elements. Start with F3, find an irreducible polynomial of…

A: Step 1:Step 2: Step 3: Step 4:

Q: Subject: Abstract AlgebraPlease help me out with these problems through proofing. There are two…

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Q: (a) Find a u, A E Z such that µ12 + 135 = 1. Show your work.. (b) Find an integer X such that X = 8…

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Q: How many elements of order 2 does the symmetric group S4 contain?

A: We know that S4 is symmetric group of all permutations of 4 elements. S4 has 4!=24 elements. To find…

Q: Verify that each of the sets {1,3,7,9,11,13,17,19}, {1,3,7,9}, {1,9,13,17} forms a group under…

A: We can verify if each set forms a group under multiplication modulo 20 by checking the following…

Q: 4. (i) Show that 2 is a primitive root mod 112 = 121. Hint: Use the criterion given by Lemma 6.4 and…

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Q: 1. Let p be a prime number. Find the number of generators of the cyclic group Z where r is an…

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Q: Prove that if n is not prime, then {1, 2, 3,..., n-1} is not a group under multiplication mod n.

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Q: 8. Compute the following. (a) (17 mod 6) (b) (256 mod 31) (c) (-2245 mod 7)

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Q: 16. Let Q = {x € Q 1 x # 0} denote the multiplicative group of non-zero rationals. Determine whether…

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…