D3, S ymmeteies of an Equillateral triang le 4123) (34 (23) (13) 42) )= Po fo M2 M3 M. C123)=p P My Po (23) = My MI Po C13) = 42 (12) = Mg My fo Find the cube of each element in S3 = D3, i.e. for each x ED 3, find %3D the element x in D3. P.

Q: QUESTION 4 a) Find the order of each element in Z15/(5). b) Find n such the this group is isomorphic…

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Q: (2) Chapter 6, exercise 7.11: Prove that the only subgroup of order 12 of the symmetric group S₁ is…

A: We know that A4 is subgroup S4 of order 12  We have to show that A4 is only subgroup of S4 of order…

Q: Chapter 2, exercise 11.6: Let G be a group that contains normal subgroups of orders 3 and 5,…

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Q: Show that U(8) and U(10) are not isomorphic.

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Q: 27. Payton ran l miles on Monday, 2 miles on 7. miles on Wednesday. What is the 8. Tuesday, and 1-…

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Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: Chapter 2, exercise 11.3: Prove that the product of two infinite cyclic groups is not cyclic.

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Q: 5. ts) Let T(x) be a linear transformation. Suppose the following ma- trix is a row echelon form of…

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Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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Q: 30. (Remainder Theorem) Let f(x) e F[x] where F is a field, and let a E F. Show that the remainder…

A: Here we have given that fx∈Fx, where F is a field and α∈F. it is required to show that the remainder…

Q: Find the order of each element in Z12-

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Q: QUESTION 5 Find the inverse of each element in (U (14), 14).

A: The set Un is the set of all integers greater than 1 that are coprime with n. The number of elements…

Q: The population of a community is known to increase at a rate proportional to the number of people…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: QUESTION 7 Show that the special linear group, SL(2, R) is non -Abelian.

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Q: QUESTION 3 Find the oder of the indicated element in indicated quotient group. a) 2 + (8)in Z12/ (8)…

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Q: Exercise 2.114 Consider the ring Q[/m] of Example 2.12. Now assume for this exercise that m is not a…

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Q: 210. 153 How many abelian groups of order n = Explain your reasoning. You may assume that the data…

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Q: Chapter 6, exercise 4.2(b): List the proper normal subgroups N of the dihedral group D₁ and identify…

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Q: Find all possible homomorphisms form Z10 → Z4. Then find the kernel of each of these homomorphisms.

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Q: (7) The dihedral group is the subgroup D4 of S4 of order 8 consisting of permutations induced by…

A: Solution 7: By Lagrange's Theorem, the possible orders are 1,2,4 and 8 The only subgroup of order…

Q: Italian Algebra

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Q: 1- Let G = (U15, ×), and let N = , the subspace generated by 2. List all the right cosets of N, and…

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Q: subgroups into subgroups. 46/If there is an a e G such that (a) = G, then G' is cyclic. %3D G then

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Q: hese problems let R be any ring and Mand N R- orem A: If ƒ :M → N is an R- module epimom Kerf≈ N…

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Q: 3. Let G be a finite group of order |G| = n. Prove that G is isomorphic to a subgroup of Sym(G)…

A: The objective is to prove that G is isomorphic to a subgroup of symmetric group.

Q: Without using the structure theorem for finite abelian groups, prove that a finite abelian group has…

A: Given: Let G be a finite abelian group. To prove: G has an element of order p, for every prime…

Q: QUESTION 15 a) How many elements of order 2 are there in S5? b) How many elements of order 2 are…

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Q: QUESTION 9 Find up to isomorphism all Abelian groups of order 36.

A: We have to find the all abelian groups of order 36 up to isomorphism.

Q: 2. Let p and q be two distinct prime numbers. Compute the order of U (pq). Use this result to…

A: We have to solve given problem:

Q: 7- Let G = U15, and let K = {1, 4}, then K is a subgroup of G. List the elements of G/K, and find…

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Q: (a) R=Z, I = 35Z.

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 4.2. An nth root of unity is a complex number z such that z" = 1. (a) Prove that the nth roots of…

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Q: UESTION5 Show that U(9)is cyclic and find all the generators of U(9).

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Q: Let S be a subring of the ring R. Thus, by definition (S, +) is an abelian group. Let 0s denote the…

A: We will take an arbitrary element from S and show that the identity property holds for both 0s…

Q: (b) Consider f(x) = 2x³ + 3x² + 4 in Zs [], and the evaluation homomorphism 2 [2]: Zs [2] → Zg. (i)…

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Q: (3) Chapter 2, exercise 8.5: A finite group contains an element x of order 10 and also an element Y…

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Q: Exercise 2.46.1 Let R be a commutative ring. Suppose that every nonzero element in R is invertible.…

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Q: 28) Let N be a finite subgroup of a group G and assume N = for some subset S of G. Prove that an…

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Q: Use the left regular representation of the quaternion group Q8 to produce two elements of Sg which…

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Q: QUESTION 6 Show that U(14) and U(18) are isomorphic..

A: We have to show that the group U(14) and U(18) are isomorphic.