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хоол одоо ч олород ото) 4. Alt(4) is the alternating group on 4 points. (a) (3...
Exercise 3 1- What are the symmetry elements of the NH molecule whose four atoms occupy the corne...
Exercise 3 1- What are the symmetry elements of the NH molecule whose four atoms occupy the corners of a regular tetrahedron. Molecular geometry of NH 2- Make the stereographic projection of symmetry elements and equivalent positions generated by its operations 3- Give the elements of the corresponding point group. Specify its order and its international notation. 4- Establish the multiplication table of this group specify its rank. 5- Is the group Abelian? Describe all subgroups of this group. Afif...
4. (a) Define when two elements of a group are conjugate to each other. State and de- duce the cl...
Please show all steps clearly.
4. (a) Define when two elements of a group are conjugate to each other. State and de- duce the class equation using the decomposition of a group in conjugacy classes (b) Let G be a finite group and p a prime number such that p divides G. Prove that there is a subgroup H of G such that |H p. (c) Let p be a prime number. Prove that any positive integer n, any group...
4. If G is a group, then it acts on itself by conjugation: If we let...
4. If G is a group, then it acts on itself by conjugation: If we let X = G (to make the ideas clearer), then the action is Gx X = (g, x) H+ 5-1xg E G. Equivalence classes of G under this action are usually called conjugacy classes. (a) If geG, what does it mean for x E X to be fixed by g under this action? (b) If x E X , what is the isotropy subgroup Gx...
Identifiy S3 with the group of S4 to 4 consisting of the permutations of (1,2,3,4 ) that maps a) ...
Identifiy S3 with the group of S4 to 4 consisting of the permutations of (1,2,3,4 ) that maps a) Write down the elements of a subgroup H of S4 that is a conjugate of Ss but not S3 itself. (Hint: any such H wl have 6 elements) (b) How many subgroups of Sa are conjugates of Ss (including Ss itself)? (c)Let H be a subgroup of a group G. Show that Nc(H), the normalizer of H in G (d) What...
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element...
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element (1,2,3,4,5,6) is a cyclic group of order 6. b. Show that the subgroup generated by the element (1,3, 4, 5, 6, 7) is a cyclic group of order 6. c. Show that the subgroup generated by the element (1,2,3) is a cyclic group of order 3. d. Show that the subgroup generated by the element (6, 7) is a cyclic group of order 2....
4. (a) (3 points) List all the subgroups of the symmetric group S3. (b) (4 points)...
4. (a) (3 points) List all the subgroups of the symmetric group S3. (b) (4 points) List all the normal subgroups of Sz. (c) (3 points) Show that the quotient of S3 by any nontrivial normal subgroup is a cyclic group.
6. 10 Suppose a cubic polynomial y2does through the points ( i) fori-1,2,3, 4, where i j for i,j ...
6. 10 Suppose a cubic polynomial y2does through the points ( i) fori-1,2,3, 4, where i j for i,j 1,2,3, 4 and i j a) 2 Find the system of equations that determines the coefficients a, b, c and d (b) |6 Find the determinant of the coefficiant matrix using row operations, and show that the coefficient matrix is invertible. Note that you will receive no mark if you compute the determinant using cofactor expansion. (c) [2| Is it possible...
For the permutation group of 4 elements (S4) - 1. What are its classes also find...
For the permutation group of 4 elements (S4) - 1. What are its classes also find the order of each class 2.Write down the dimensions of all the irreducible representations
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
11. (10 points) - MATLAB ce of the alternating harmonic series, shown below: 1-2?--+ - 3...
11. (10 points) - MATLAB ce of the alternating harmonic series, shown below: 1-2?--+ - 3 4 5 Use a loop to determine when the series has converged within a tolerance of 0.001. The animation should show how the value of the series changes as each successive term is added, resulting in a final
Exercise 3 1- What are the symmetry elements of the NH molecule whose four atoms occupy the corners of a regular tetrahedron. Molecular geometry of NH 2- Make the stereographic projection of symmetry elements and equivalent positions generated by its operations 3- Give the elements of the corresponding point group. Specify its order and its international notation. 4- Establish the multiplication table of this group specify its rank. 5- Is the group Abelian? Describe all subgroups of this group. Afif...
Please show all steps clearly.
4. (a) Define when two elements of a group are conjugate to each other. State and de- duce the class equation using the decomposition of a group in conjugacy classes (b) Let G be a finite group and p a prime number such that p divides G. Prove that there is a subgroup H of G such that |H p. (c) Let p be a prime number. Prove that any positive integer n, any group...
4. If G is a group, then it acts on itself by conjugation: If we let X = G (to make the ideas clearer), then the action is Gx X = (g, x) H+ 5-1xg E G. Equivalence classes of G under this action are usually called conjugacy classes. (a) If geG, what does it mean for x E X to be fixed by g under this action? (b) If x E X , what is the isotropy subgroup Gx...
Identifiy S3 with the group of S4 to 4 consisting of the permutations of (1,2,3,4 ) that maps a) Write down the elements of a subgroup H of S4 that is a conjugate of Ss but not S3 itself. (Hint: any such H wl have 6 elements) (b) How many subgroups of Sa are conjugates of Ss (including Ss itself)? (c)Let H be a subgroup of a group G. Show that Nc(H), the normalizer of H in G (d) What...
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element (1,2,3,4,5,6) is a cyclic group of order 6. b. Show that the subgroup generated by the element (1,3, 4, 5, 6, 7) is a cyclic group of order 6. c. Show that the subgroup generated by the element (1,2,3) is a cyclic group of order 3. d. Show that the subgroup generated by the element (6, 7) is a cyclic group of order 2....
4. (a) (3 points) List all the subgroups of the symmetric group S3. (b) (4 points) List all the normal subgroups of Sz. (c) (3 points) Show that the quotient of S3 by any nontrivial normal subgroup is a cyclic group.
6. 10 Suppose a cubic polynomial y2does through the points ( i) fori-1,2,3, 4, where i j for i,j 1,2,3, 4 and i j a) 2 Find the system of equations that determines the coefficients a, b, c and d (b) |6 Find the determinant of the coefficiant matrix using row operations, and show that the coefficient matrix is invertible. Note that you will receive no mark if you compute the determinant using cofactor expansion. (c) [2| Is it possible...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
11. (10 points) - MATLAB ce of the alternating harmonic series, shown below: 1-2?--+ - 3 4 5 Use a loop to determine when the series has converged within a tolerance of 0.001. The animation should show how the value of the series changes as each successive term is added, resulting in a final
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