3.1 Solutions
36. For each binary operation dened on a set below, determine whether or not give a group structure on the
set. If it is not a group, say which axioms fail to hold.
We evidently get to assume that each of the operations is a binary operation
3.5
3.5
J.A.Beachy
1
Cyclic Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
21. Show that the three groups Z6 , Z , and Z are isomorphic to each other.
9
18
Solution: First, we have |Z | = 6, and |
3.6
3.6
J.A.Beachy
1
Permutation Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
28. In the dihedral group Dn = cfw_ai bj | 0 i < n, 0 j < 2 with o(a) = n, o(b) = 2,
and ba = a1 b, show that bai =
3.5
3.5
J.A.Beachy
1
Cyclic Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
We began our study of abstract algebra very concretely, by looking at the group Z of
integers, and the related groups Zn
3.3
3.3
J.A.Beachy
1
Constructing Examples
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
The most important result in this section is Proposition 3.3.7, which shows that the
set of all invertible n n ma
3.4
3.4
J.A.Beachy
1
Isomorphisms
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
A one-to-one correspondence : G1 G2 between groups G1 and G2 is called a group
isomorphism if (ab) = (a)(b) for all a, b G
3.6
3.6
J.A.Beachy
1
Permutation Groups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
As with the previous section, this section revisits the roots of group theory that we
began to study in an earlier c
3.2
3.2
J.A.Beachy
1
Subgroups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
28. In Zn , show that if gcd(a, n) = d, then [a]n = [d]n .
Note: This result is very useful when you are trying to nd cyclic
3.2
3.2
J.A.Beachy
1
Subgroups
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
Many times a group is dened by looking at a subset of a known group. If the subset
is a group in its own right, using the sam
3.1
3.1
J.A.Beachy
1
Denition of a Group
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
25. Use the dot product to dene a multiplication on R3 . Does this make R3 into a
group?
Solution: The dot product
MATH 420
FINAL EXAM
May 5, 2010
Prof. John Beachy
No calculators.
Each question is worth 25 points.
Show all necessary computations.
1. Find the multiplicative inverse of [38]83 in Z .
83
2. Solve the following system of congruences:
2x 5 (mod 7)
3x 4 (mo
MATH 420
Prof. John Beachy
TEST III
April 16, 2010
Each problem is worth 20 points. Show all of the work necessary to justify your answers.
You are not allowed to use a calculator.
1. (a) State the denition of a group.
(b) State the denition of an abelian
MATH 420
Prof. John Beachy
1. (25 pts) Let =
TEST II
123456789
147596328
March 19, 2010
and =
123456789
437586219
(a) Write each of , , , , and 1 as a product of disjoint cycles.
(b) Find the order of each of , , , , and 1 .
(c) Determine whether each of
MATH 420
Prof. John Beachy
TEST II Solutions
March 26, 2010
1. (25 pts) Write each of , , , , and 1 as a product of disjoint cycles, nd their order, and determine
whether each is even or odd.
123456789
=
= (2, 4, 5, 9, 8)(3, 7) has order lcm[5, 2] = 10 an
MATH 420
Prof. John Beachy
TEST II
October 27, 2006
1. (15 pts) Dene f : Z8 Z12 by f ([x]8 ) = [3x]12 , for all [x]8 Z8 .
(a) Show that f is a well-dened function.
Recall: you must show that if x1 x2 (mod 8), then 3x1 3x2 (mod 12).
(b) Find the image f (Z
MATH 420
Prof. John Beachy
TEST II Solutions
October 27, 2006
1. (15 pts) Dene f : Z8 Z12 by f ([x]8 ) = [3x]12 , for all [x]8 Z8 .
(a) (7 pts) Show that f is a well-dened function. Show that if x1 x2 (mod 8), then 3x1 3x2 (mod 12).
If [x1 ]8 = [x2 ]8 , t
MATH 420
Prof. John Beachy
EXAM I
Show all necessary work.
February 12, 2010
No calculators.
1. (15 pts) Find gcd(1492, 1776)
(a) by using the Euclidean algorithm;
(b) by nding the prime factorizations of 1492 and 1776.
2. (15 pts) Solve the system of con
MATH 420
Prof. John Beachy
TEST I
Show all necessary work.
June 29, 2009
No calculators.
1. (15 pts)
(a) Let a, b be nonzero integers. State the denition of gcd(a, b).
(b) Use Theorem 1.1.6 (see the statement in Problem 5) to prove that if a, b, c are non
MATH 420
Prof. John Beachy
TEST I Solutions
Show all necessary work.
June 29, 2009
No calculators.
2. (10 pts) Find gcd(323, 391) and write it as a linear combination of 323 and 391.
gcd(323, 391) = 17 = (5)(391) + (6)(323.
1
0
0
1
391
323
1
0
1
1
68
323
3.4
3.4
J.A.Beachy
1
Isomorphisms
from A Study Guide for Beginners by J.A.Beachy,
a supplement to Abstract Algebra by Beachy / Blair
29. Show that Z is isomorphic to Z16 .
17
Comment: The introduction to Section 3.2 in the Study Guide shows that the eleme