76 CHAPTER 6. COSETS AND LAGRANGES THEOREM
Remark 6. 14 (The converse of Lagranges Theorem is false). The group
A4 has order 12; however, it can be shown that it does not possess a
subgroup of order 6
7.1. PRIVATE KEY CRYPTOGRAPHY 81
apply to the message some function which is kept secret, say f. This
function will yield an encrypted message. Given the encrypted form of
the message, we can recover
5.3. EXERCISES 69
Proposition 5.27. The group of rigid motions of o. eabe contains 24
elements.
Theorem 5.28. The group of rigid motions of a cube is 84.
PROOF. From Proposition 5.27, we already know
66 CHAPTER 5. PERMUTATION GROUPS
5 .2 Dihedral Groups
Another special type of permutation group is the dihedral group. Recall
the symmetry group of an equilateral triangle in Chapter 3. Such groups
co
5.3. EXERCISES 71
20. Let cfw_I 6 Sn be a cycle. Prove that 0' can be written as the product
of at most in. 1 transpositions.
21. Let 0' E S. If (I is not a cycle, prove that cfw_T can be written as t
62 CHAPTER 5. PEWUTATION GROUPS
Proposition 5.8. Let er and T be two disjoint cycles in 3);. Then
UT = TU.
PROOF. Let o : (o1,a2, . . . ,ak) and T = (b1,b2, . . .,l)g). We must show
that JTCII) = TJ($
6.2. LAGRAN GE 8 THEOREM 75
Example 6.7. Suppose that G = 83, H = cfw_(1), (123), (132), and K =
cfw_(1), (12). Then [G : H] = 2 and [G : K] = 3.
Theorem 6.8. Let H be a subgroup of a group G. The num
78 CHAPTER 6. COSETS AND LAGRANGES THEOREM
6.4 Exercises
1. Suppose that G is a nite group with an element 9 of order 5 and an
element h of order 7. Why must |G| 2 35?
2. Suppose that G is a nite grou
6.4. EXERCISES 79
18. If [G : H] = 2, prove that gH = Hg.
19. Let H and K be subgroups of a group G. Prove that 91:? 0 9K is a
coset of H m K in G.
20. Let H and K be subgroups of a group G. Dene a re
5.2. DIHEDRAL GROUPS 67
#1:
1 1
6 2 2 6
5 3 3 5
4 4
1 1
5 2 2 5
>
4 3 3 4
Figure 5.22: Types of reections of a regular agon
Theorem 5.23. The group D, n 2 3, consists of all products of the
two elemen
68 CHAPTER 5. PERMUTATION GROUPS
rotations are
r = (1234)
r2 = (13)(24)
r3 = (1432)
P4 = (1)
and the reections are
$1 = (24)
82 = (13).
The order of D4 is 8. The remaining two elements are
r31 = (12)(
5.1. DEFINITIONS AND NOTATION 63
Using cycle notation, we can write
a = (1624)
r = (13)(456)
or : (135)(245]
m = (143)(256).
Remark 5.11. From this point forward we will nd it convenient to use
cycle
74 CHAPTER 6. COSETS AND LAGRANGES THEOREM
The right eosets of H are exactly the same as the left oosets:
H(1) : H(123) : H(132) : cfw_(1), (123), (132)
H(12) = H(13) = H(23) = cfw_(12), (13), (23).
I
72 CHAPTER 5. PEWUTATION GROUPS
(d) If OMr Oyl. 7E (ll, prove that (93Mr = (93,30. The orbits under
a permutation or are the equivalence classes corresponding to the
equivalence relation w.
(e) A subg
6.3. FERMATS AND EULERS THEOREMS 77
6.3 Fermats and Eulers Theorems
The Enter (trfunction is the map (,6 : N > N dened by (Mn) = 1 for
n : 1, and, for n 2: 1, q5(n) is the number of positive integers
Net Change in account
$0.00 $3,333.34
$(5.00) $3,000.00
$2.50 $3,500.00
$10.00
$4,000.00
$(20.00) $2,000.00
In paraenthesis indicates the number is negative
#
Questions
1)
Model
(0.015An)-50
2)
Equili
Equilibrium values of difference equations are when the dependent variables is equal to a
point where the equation an = f(an) equals zero. For difference equations a stable point is
defined as a point
Time since Blue crabs (lb)
1940 by 5 yrs
in 10^4 lb
model
x
y
Y
0
10
10
1
85
84.9957
2
133
132.9528
3
250
249.7693
4
300
299.2752
5
370
368.2125
6
440
436.1992
7
-739.3227
3.5
282.7047
2.5
194.99
500
Equilibrium values for difference equations are points such as t = 1 where the function
neither increases nor decreases. In difference equations the term stable is very ambiguous being
that the functi
CHOOSE FOUR OF THE SEVEN QUESTIONS BELOW:
1. Answer the Socratic question, "How should one live?"
2. Describe the four major approaches to moral differences outlined at the
beginning of Chapter 3. Whi
QUIZ 5 / FINAL
STUDY GUIDE
Below is a list of terms/concepts/theories to study for the final. The list is in the order of how they
were presented in the power points. They may be in a random order on
Introduction to
Cryptography
Cryptography is the study of sending and receiving secret messages. The
aim of cryptography is to send messages across a channel so that only
the intended recipient of t
Cosets and Lagranges
Theorem
Lagranges Theorem, one of the most important results in nite group
theory, states that the order of a subgroup must divide the order of the
group. This theorem provides
64 CHAPTER 5. PEWUTATION GROUPS
where a, b, c, and d are distinct.
The rst equation sirnpl,r says that a transposition is its own inverse.
If this case occurs, delete r,._1r,. from the product to obta