A 2-dimensional 4-fold rotation (for example, a square) symmetry forms a group G with 4 operations: E- Identity C4 – Rotation (about the center) by n/2 C2- Rotation by n Ca – Rotation by 3 n/2. 1) Write out the complete multiplication table. 2) Find out all the subgroups from it. 3) Give a 2x2 matrix representation of the group (for all 4 members of the group). 4) Is this group orthogonal?

Q: (а) Show that for any group G, Z(G) = N Z(x). %3D xeG (b) Find the degree of Q(5, 17) over Q. Give…

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Q: 4) Find the smallest integer n such that an+3 = a. 5) Prove that a is an odd permutation. 6) Let ß =…

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Q: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

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Q: Chapter 2, exercise 6.4: Prove that in a group, the products ab and ba are conjugate elements.

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Q: X={1,2,0,7,5}. (a) Find all possible permutations and show that its form a group. (b) Find its…

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Q: 6) Chapter 7, exercise 4.1: The icosahedral group operates on the set of five inscribed cubes in the…

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Q: For a cyclic group of order 420, the number of generators are, (a) 66 (b) 76 (c) 86 (d) 96

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Q: Chapter 6, exercise 4.1: Write the product x²yx-¹y-¹x³y³ in the form x¹y¹ in the dihedral group Dn.

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Q: Let G = : a – b = c – d, a,b, c, d E R Show that G is a group under (the usual) matrix addition.

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Q: The group of of a regular n- sided polygon is called the dihedral group of order 2n.

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Q: 2. a) The operation in a finite group can be specified by writing down a table. Write the table for…

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Q: Let Rn be the group of rotations of a regular pentagon, and let r be counterclockwise rotation by…

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Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

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Q: Show that O(n), the set of all orthogonal n × n matrices is a group under matrix multiplication

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Q: Write the three-dimensional matrices that represent each of the operations of the point group D2n…

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Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

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Q: The n-th dihedral group Dn is the group of symmetries on a regular n-gon. The group of symmetries on…

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Q: Find order of all the elements of the Group = {1,-1, i, -i}. The binary operator * is defined as a *…

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Q: Find the order of G. Find the order of i and -1.

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Q: Q 5 The order of the element (123) of the group (S3,0) is (a) 1 (b) 2 (c) 3 (d) 00 Q 6 The order of…

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Q: Let Rn be the group of rotations of a regular pentagon, and let r be counterclockwise rotation by…

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Q: Develop the irreducible (2×2) matrix representations of the group of rotations (including those that…

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