Mathematics 366
Homework (due Nov. 11)
A. Hulpke
56) 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 .
57) 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 letters 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 M
G
M
N
O
P M N
R
N
O
P
R
N
G
R
a) What is the identity element of the group? Why?
b) What is the Order of the Phoenix?
58∗ ) 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?
59∗ ) Is U(8) a subgroup of U(32)? Explain. (Hint: What is 5 ⋅ 5 in both groups?)
60) Let G = S 5 . Which of the following subsets are subgroups?
a) U 1 = {p ∈ S 5 ∣ 1 p = 1}
b) U 2 = {p ∈ S 5 ∣ 1 p = 2}
c) U 3 = {p ∈ S 5 ∣ ∣p∣ ≤ 2}
61)
Let G be a group, H ≤ G and x ∈ G. We define a set H x by:
H x ∶= xHx −1 = {xhx −1 ∣ h ∈ H}
Show that H x ≤ G.
Problems marked with a ∗ are bonus problems for extra credit.
Sample Midterm Problems
The second midterm will be on November 11. Rules are as in midterm 1 (I.e. you may bring a single page of unrestricted
notes. It must be handwritten by yourself and carry your name. You may only consult your own notes during the exam.)
This is a set of problems which are (or are similar to) previous semesters midterm problems. The midterm problems
will be chosen to be similar to these or to homework problems (though the total number of midterm problems will be
smaller).
1) Let G = S 6 the group of all permutations on the points 1 to 6 with composition as operation.
Which of the following sets H ⊂ G are subgroups? (Prove that they are subgroups or give a counterexample. You do not
need to show that H ⊂ G.)
a) H 1 = {a ∈ G ∣ a(5) = 5}.
b) H 2 = {a ∈ G ∣ a(5) = 4}.
2)
Show that S n is not abelian for n ≥ 3.
3) 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
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
4) Let G = Q∗ ∶= Q − {0} with the ordinary multiplication as operation. Which of the following sets H ⊂ G are
subgroups? (Prove that they are subgroups or give a counterexample.)
a) H 1 = {a ∈ Q∗ ∣ a > 0}.
b) H 2 = {a ∈ Q∗ ∣ a < 0}.
c) H 3 = {a ∈ Q∗ ∣ a = dn for some n, d ∈ Z with d odd}.
5) Determine the elements of U(12) and their orders. Is U(12) cyclic?
6) Let R = Q[x] and let A = { f ∈ R ∣ f (1) = f (2) = 0}.
a) Show that A is an ideal in R .
b) Is A a prime ideal in R?
c) Is A a maximal ideal in R?
7) Let R = Z[x] the ring of polynomials with integer coefficients and I = ⟨2, x 2 − 1⟩R ⊲ I.
a) Calculate (I + x) ⋅ (I + (x + 1)) in R/I.
b) Show that R/I is not an integral domain by finding explicitly a zero divisor.
c) Is I a maximal ideal of R?
8)
Let φ∶ R → S be a ring homomorphism and A ⊲ S. Let B = {r ∈ R ∣ φ(r) ∈ A}. Show that B ⊲ R.
9) Construct (you do not need to write down the full addition and multiplication tables, a description of how to
construct it will be sufficient) a field with 16 elements and show that it is a field. (Hint: Consider F2 [x]/ ⟨p(x)⟩ for a
suitable polynomial of degree 4.)
10) Let F be a field and A ∈ M n (F) an n × n matrix. Show that the map θ∶ F[x] → M n (F), p(x) ↦ p(A) (the
polynomial evaluated at the matrix) is a ring homomorphism. Conclude that there is a polynomial m A (x) ∈ F[x], such
that every polynomial p(x) ∈ F[x] for which p(A) = 0n×n is a multiple of m A . (m A is called the minimal polynomial
of A.)