Exercises 1. Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? a. Exercise 1 b. Exercise 2 c. Exercise 3 d. Exercise 4 e. Exercise 5 f. Exercise 6

Q: Draw the S, Z and PI shaped membership functions to represent the young, middle-aged and old persons…

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Q: - Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,

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Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

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Q: Is {0, 2, 4, 6} a subgroup of Zg? Justify your answer.

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Q: Answer the followings: 1. nZ is a normal subgroup of Z. a. True b. False Compute (1 3) (1 2) (1 3)−¹…

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Q: 3.40. Let G be an abelian group and n e N. Let H = {a e G : a" = e} and K = {a" : a e G}. Show that…

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Q: The normalizer N(a) of a∈G is a subgroup of G.

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Q: When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does…

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Q: r2r2 = (Round to two decimal places) Interpret r2r2 :

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Q: Let n E Z be an integer number and let H = (n) = {rn,r E Z} i.e\ H = (n) = {.., -3n, –2n, -n, 0, n,…

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Q: Q2.3 Question 1c Let G = Są and let H = {o € S4 | o (2) = 2}. Then %3D O H is not a subgroup in G O…

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Q: Abstract algebra higher level 1. If A and B are both normal in G. then 1. AB is a subgroup of G 2.…

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Q: 3. am e H for every a E G. Let H be a normal subgroup of a group G, and let m = (G : H). Show that

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Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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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: Exercise 6.3.12. Suppose G is a group, a, b e G such that gcd(la\, [b|) = 1. Prove (a) n (b) = {e}.

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Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

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Q: Explain the following statement "If G is a group an a E G then o(a) = | |." 31. %3D

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Q: Class Intervals Class Boundaries 36-40 35.5-40.5 31-35 30.5-35.5 26-30 25.5-30.5 21-25 20.5-25.5…

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Q: Class Intervals Class Boundaries 36-40 35.5-40.5 31-35 30.5-35.5 26-30 25.5-30.5 21-25 20.5-25.5…

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Q: et H < G. Then Gµ = {g € G | gHg¬ = H} is a subgroup of G that has H has a -ormal subgroup. %3D

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Q: Let G = U56 and define two subgroups H₁ and H₂ of G by := H₁ := Calculate H₁ H₂|. {1, 5, 9, 13, 25,…

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Q: Let G be a group and a 2 G. Then the set = {a" : ne Z}is a subgroup of G. 48.

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Q: Let G = S4 and N {e, (1, 2) (3, 4), (1, 3) (2, 4), (1, 4) (2,3)}. Show that N is a normal subgroup.…

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A: (c) Suppose the given structure {G = {a, b, c, d}, ∘} forms a group.       Now, a ∘ a = a. . . . (1)…

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A: given  G=c,d|c4=d2=cd2=e be a group and the subgroup H=c|c4=eof group G claim- find the centralizer…