MATH 4361 Exam 3 (Version 1) Solutions July 10, 2007 S. F. Ellermeyer Name Instructions. Please be detailed in your solutions and proofs. Write in complete sentences! The harder that I have to work to try to interpret what you are trying to say, the less
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 5
Problem 6.1. Let (2) and (10) be subgroups of Z under addition. Note (10) < (2) <
Z .
(a) List (or describe the elements of) the left cosets of (10) in (2).
If G = < 2 > and H = < 10 >,
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 2
Problem 2.1. Which has larger cardinality among the following pairs?
(a) Z23 or Z24
|Z23| = 23 and | Z 24 |= 24
Z 23 < Z 24
(b) U (23) or U (24)?
U(23) =
( 23 )= (231 230) = (23 1) = 22
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 4
Problem 4.1. Complete the following for the group Z24.
(a) List all subgroups of Z24 (and write out the elements of each).
Z
24
Order
Order
Order
Order
Order
Order
Order
Order
= cfw_1, 2
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 3
Problem 3.1. Find the order of each element, all subgroups, and draw the subgroup lattice for
each of the following groups:
(a)
Z5
|0| = 1, |1| = 5, |2| = 5, |3| = 5, |4| = 5
< 0 > = cfw
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 5
Problem 5.1. Decide for each pair, if they are isomorphic:
(a) U (6) and Z7 Not isomorphic: |U(6)| = 2 |Z7| = 7 thus, no possible bijection can exist.
(b) U (7) and Z6 Isomorphic: U(7) =
Faith Eurie-Turner
Math 4361 Modern Algebra I
Summer 2016 Homework 1
Problem 1.1. For each property below, give a binary operation on a set S = cfw_a, b, c using an arithmetic table.
Also, supply proof if you claim is associative or commutative. Suppl