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
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
Question 1(a), Question 2 second part, Question 9
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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