Let G be a finhte cyclic geupnd a be an clem ent of G of order 8 then the order of G would be O 18 24 2

Q: 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

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Q: For any set S there existe eS such that a*e=e*a=a, va e S then e is called O a. a neutral element O…

A: Let me discuss this problem in step 2-->

Q: 1. (p.209, #16) Define a relation R on Z by declaring that xRy if and only if x? = y? (mod 4). Prove…

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Q: Let m be an integer with m > 1. Show that the relation R = {(a, b) | a ≡ b (mod m)} is an…

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Q: 4. Prove or disprove the following statement. U(8) is cyclic. [from #1, 4.5]

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Q: The relation R on the set A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is defined by R= {(x, y) E AXA: x…

A: Here according to the definition,       R = {(x,y) : x = y mod 3} Then if y = 7, then x must be 7…

Q: 7. Prove the following: Given: PI || NG, PI = NG Prove: APIN ANGP G N P.

A: Given : PI¯∥NG¯, PI¯≅NG¯ Consider the given figure, in which given situation is shown. Here…

Q: Given a relation R on Z defined by mRn 7(m-n) for all m, n e Z. Find the equivalent class determined…

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Q: For any set S there exist be S such that a*b = e for some a e S then bis called which element of a?…

A: A unit in a ring R is the element u in R that has a multiplicative inverse in R: an element v in R…

Q: From the following options choose the appropriate values of a, b and m satisfying the given…

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Q: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

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Q: Prove or disprove: If H and G/H are cyclic, then G is cyclic.

A: Important key points: 1. If order of a group is prime, then G is cyclic.2. S3, the set of all…

Q: Let H x K = { (h,k) | h in H, k in K } such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: The objective is to find the order of Z_2 x Z_2.

Q: Given a relation R on Z defined by mRn ⇔ 3|(m −n) for all m, n ∈ Z. Find the equivalent class…

A: We have the relation defined by mRn⇔3/m-n Now ,we need to find the equivalence class determined by…

Q: 3. Show that the following refinement is correct (for any g, S, and S,): {-g}; if g then S, else S2…

A: Given refinement is: ~g; if g then S1 else S2. Here ~g implies the negation part of g. That is the…

Q: Q) Make up a Kelly table for set G= [1,-1, i,-i} with respect to multiptication and then explain…

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Q: Let H x K = { (h,k) |h in H, k in K} such that (h1,k1) + (h2,k2) = (h1 + h2, k1 + k2), for all h1,…

A: here ℤ2=0,1

Q: 7. Recall that, for n e Z, n > 0, there is a relation on Z called congruence mod n. The b (i.e.…

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

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Q: 3. Let S be the congruent modulo 2 relation on Z, that is m S n→2|(m-n). Show the relation S is: (a)…

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Q: 11 Let S be the congruent modulo 5 relation on Z , that is m S n → 5|(m – n) . Show the relation S…

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Q: (a) Evaluate I = SoSi L' dxdydz. (b) In how many possible orders can you calculate I? Do all give…

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Q: For a set S = {a1,..., an} consider the cycle f = (a1 ... a, ). What properties does the cycle f…

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Q: 11. E is the relation defined on Z as follows: for all m,n e Z, mEn → 4|(m-n). a) Prove that E is…

A: Given a relation E on Z as follows: for all m,n∈Z, mEn⇔4m-n.

Q: Show that for every x E R, there exists an n EN such that <x<n.

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Q: 4. Let r E R" be such that r = (E1,... , En). Show that the norms ...+ En and = V are equivalent so…

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Q: Prove if a e G, o(a) =n, then a = reN GCD(n,r)'

A: Given below the detailed solution

Q: b( Let S be the relation defined on P ((1, 2, 3, 4}) defined by X SY if and only if IX| = |Y| (mod…

A: The complete solutions are given below

Q: 11. Which of the following is a subsemigroup of (Q,*) where a*b = ab/5? A. (Z, *) C. (Q+, *) B. (Z+,…

A: Given that Q,* is a semigroup where a*b=ab5∀a,b∈Q

Q: 2) Let m +0 be an integer. Show that Rm := { (a, b) E Z × Z|m divides a – b} is an equivalence…

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Q: 2. Suppose N= {1,2,3...} is the universal set and A = {n :ns7} B = {n : 3s ns 10} %3D C =…

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Q: Let A = {1, 2, 3, 4} and consider the relation R on A given by aRb if and only if a−b ≡ 0 mod 4, or…

A: Given: The set A=1, 2, 3, 4, and the relation R:A→A as  R=a,b: a-b≡0mod4 or a-b≡1mod4 or a-b≡-1mod4…

Q: 3 S how that Set M with n elements. a where n is a elem ent of ne tural numbers (ne IN. has 2…

A: Number of elements in M is n. To prove power set of M has 2n elements. We know that Power set of M…

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

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Q: In proving that G/N is a group where do we first use the fact that N Is ha Select one: a. The invers…

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Q: 15.)Which of the following subsets of C (set of com- plex numbers) is not a subgroup of (C, +)? The…

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Q: Let S be the congruent modulo 7 relation on Z, that is m Sn 7|(m-n). Show the relation S is: (a)…

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Q: equivalence classes in the relation R = {(a,b) : a,b E Z, a = b(mod4} is IS

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Q: (S, *) is said to be algebraic binary structure if O a. * is associative on s O b. * is function on…

A: To find - S, * is said to be algebraic binary structure if ….. ?     Definition used - Algebraic…

Q: Let (a) H = {(), (1,2), (3 4 5 6), (35) 46), (36 54), (12) ( 3 4 5 6) (12) 35)(46), (12) (3654) 3 K…

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Q: Let f and g be permutations on the set {1, 2, 3, 4, 5, 6, 7, 8, 9}, defined as follows f = (317)(5…

A: A. Consider the transpositions of f f=3 13 75 65 7 Since the number of transpositions of f is 4,…

Q: 1. Let p e Z be a prime number and set Z, = {" e Q : If ged(n, m) = 1, then p {m}. %3D d. Show that…

A: The answer for the above question is given below please do upvote if you like the solution thank you

Q: Prove or disprove the following statements: (a) Let a, be G such that both a & b of finite order,…

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

Q: 5. Consider the permutations f 1 2 3 4 2 1 3 4 on the set {1, 2, 3, 4). (a) Find fog and go f. (b) F…

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Q: For any set S there exist be S such that a*b = e for some a e S then bis called which element of a?…

A: we have given, For any set S there exist b∈S such that a*b=e for some a∈S where e is the identity…

Q: Let d=(1234)(3456)E S, be permutation. a 9. (i) Decidle whe ther d odd. even or Find the ader of d…

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Q: . Let S be the congruent modulo 7 relation on Z, that is m Sn 7|(m-n).Show the relation S is: (a)…

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Q: Let X = N, E = P(X) and µ be the counting measure on o-algebra E. For each n e N, let An = lim…

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Q: Let : fiaiy) = x@y? (12-x-y) the lon diti on Shon that C169ラ (hi2)t

A: Given that, fx,y=x3y212-x-y, x&gt;0, and y&gt;0. To prove, fx,y≤6912. The given function can be…