Mathematics 466
Homework (due Nov. 7)
A. Hulpke
38) Which of the following sets are groups? Either show that the axioms are fulfilled or show which axiom is violated:
a) Even integers under addition.
b) Even integers under multiplication.
c) Odd integers under addition.
d) {z ∈ C ∣ ∣z∣ = 1} under ordinary multiplication.
e) {1, 2, 3} under multiplication modulo 4.
f) {0, 1, 2, 3, 4} under multiplication modulo 5.
g) The set of all matrices in R3×3 with determinant −1 (unde r matrix multiplication).
h) The set of all matrices in R3×3 of the form
⎛ 1
⎜ 0
⎝ 0
a
1
0
b ⎞
c ⎟
1 ⎠
a, b, c ∈ R
under matrix multiplication.
i) For a given real number c, the numbers in (−c, c) with operation (this is the addition of velocities in special
relativity)
x+y
x○y=
1 + x y/c 2
Hint: You do not have to prove again statements we have proven more generally in the lecture (for example: Addition
and Multiplication of complex numbers is associative, matrix multiplication is associative), but can just refer to it.
Do not forget to check that the set is closed under the operation and that inverse elements are indeed in the set!
39)
Show that S n is not abelian for n ≥ 3.
40) Let G be a (not necessarily abelian!) group and a, b ∈ G.
a) Show that (a−1 )−1 = a. (Hint: You have to show that x = a fulfills the defining condition: xa−1 = e, a−1 x = e.)
b) Show that (ab)−1 = b −1 a −1 .
c) Give an example of a group G and elements a, b, such that (ab)−1 =/ a −1 b −1 . d) Show that ((ab)2 )−1 = b−1 a −1 b −1 a−1 .
41) Show that the following multiplication table cannot be the Cayley table for a group:
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a
b
c
d
e
a
a
b
c
d
e
b
b
a
d
e
c
c
c
d
e
b
a
d
d
e
a
c
b
e
e
c
b
a
d
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
42) a) Show that the order of a permutation p is the least common multiple of the length of its cycles.
b) What is the largest possible order of a permutation on 8 points?
43∗ ) The Irish postal service (this is a true story) used a machine (“automatic letter facer”) to turn letters in the right
position for checking and validating the stamp. For this, it uses a mechanism that can (for example by “tipping the letters
over”) perform the following transformations:
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
id
a
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
b
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
c
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
Cam the Ram
8033 Campus delivery
Fort Collins, CO 80523
The machine then uses a stamp detector (S) that indicates whether a valid stamp was found in the upper right corner of
the letter.
It applies the following checking process:
Input
- S
yes
-
?no
b
? yes
S
no ?
b
6
-
yes
6
@
R c-
Accept
yes
- c - a - S - Reject
S no
no
Apparently the people who built the machine did not realize that {id, a, b, c} form a group under composition. This
makes it possible to simplify the construction:
a) Compute the multiplication table for {id, a, b, c}.
b) Suggests a simplified construction that does fewer letter flips (so processing is faster)
c) The mechanism for the “rotation” a) is relatively complicated (and thus error prone). Suggest a simplified construction
in which no operation of type a) is needed (thus simplifying the engineering).
44∗ ) The fifth year at Hogwarts school is dominated by an algebra course. The students for example learn how to
perform group operations on a set consisting of: A Griffin, a Manticore, a Naga, an Owl, a Phoenix and a Roc. (We
will denote these beasts by their first lettes and thus have a set {G, M, N , O, P, R}.)
Alas, a miscast spell has accidentally erased most of the multiplication table on the blackboard. The only remaining bit
is:
G
G
M
N
O
P M
R
M
N
O
P
N
N
G
a) What is the identity element of the group? Why?
b) What is the Order of the Phoenix?
Problems marked with a ∗ are bonus problems for extra credit.
R
R