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1. The following is an addition table and part of the multiplication table for a ring with four elements. (a) Fill out the missing entries. (b) Is this a commutative ring? (c) Does it have a unity? (+) a b C d a bed a b c d bad c cda b d cb a (-) a b C d abcd a a a a a b C a

1. The following is an addition table and part of the multiplication table for a ring with four elements. (a) Fill out the missing entries. (b) Is this a commutative ring? (c) Does it have a unity? (+) a b C d a bed a b c d bad c cda b d cb a (-) a b C d abcd a a a a a b C a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1(a), Question 2 second part, Question 9

1. The following is an addition table and part of the multiplication table for a ring with four elements. (a) Fill out the missing entries. (b) Is this a commutative ring? (c) Does it have a unity? (+) a b C d a b a b b a cd с d cd a d c d с b b a a b C d a b cd a a a a a C a b c a a 2. State Lagrange's Theorem. Then find all subgroups of the octic group (of rigid motions of the square). 3. Give 3 distinct examples of a group of order 20160. Justify each an- swer. 4. Express the following element of S7 as a product of disjoint cycles: (4215) (3426) (5671). 5. Find the cube roots of -8i. Express them in algebraic form. 6. Use Cramer's Rule to solve the given system of equations. -8x - 4y - 2z = 64 9x+6y-z=-74 9x +9y+9z=-45 7. Find the matrix for acting on the vector space V of polynomi- als of degree 3 or less in the ordered basis B (x³, x², x, 1). - 8. Give a definition of a normal subgroup H of a group G 9. Find all cosets of the subgroup Z3 in the group Z₁. 10. Let h: G→ H be a group homomorphism. Show that the kernel of h is a subgroup of G.
Transcribed Image Text:1. The following is an addition table and part of the multiplication table for a ring with four elements. (a) Fill out the missing entries. (b) Is this a commutative ring? (c) Does it have a unity? (+) a b C d a b a b b a cd с d cd a d c d с b b a a b C d a b cd a a a a a C a b c a a 2. State Lagrange's Theorem. Then find all subgroups of the octic group (of rigid motions of the square). 3. Give 3 distinct examples of a group of order 20160. Justify each an- swer. 4. Express the following element of S7 as a product of disjoint cycles: (4215) (3426) (5671). 5. Find the cube roots of -8i. Express them in algebraic form. 6. Use Cramer's Rule to solve the given system of equations. -8x - 4y - 2z = 64 9x+6y-z=-74 9x +9y+9z=-45 7. Find the matrix for acting on the vector space V of polynomi- als of degree 3 or less in the ordered basis B (x³, x², x, 1). - 8. Give a definition of a normal subgroup H of a group G 9. Find all cosets of the subgroup Z3 in the group Z₁. 10. Let h: G→ H be a group homomorphism. Show that the kernel of h is a subgroup of G.
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