Exercises

Question 1

1.1
Using Euclid’s Algorithm, compute multiplicative inverse of 11 in Z/101Z.

Question 2

2.1
If #G = p prime, show that G is cyclic.

Question 3

2.3
Let G be abelian with square-free order, show that G is cyclic.

Question 4

3.2
Let V_4 = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} < S_n.

(ii) Let o be a 3-cycle. Show that A_4 / V_4 = <oV_4>/</oV_4>

(iii) Show that S_4 / V_4 is a non-cyclic group of order 6.

Question 5

3.4
G = <g> order m, H = <h> order n. Show that \phi(g^i) = h^i is a well-defined homomorphism iff. n|m.</h></g>

Question 6

4.5
H \subset G, K < G, \phi: H x K -> HK, \phi(h, k) = hk.

(ii) Show that \phi is bijective iff. H \cap K = {1}.

Question 7

5.1
Let G group and g \in G have order n. Show that g^m has order n/gcd(m, n).

Question 8

5.2.i
Let n >= 3, determine the elements of order 2 in D_2n.

Question 9

5.2.ii
Determine the elements of order 2 in Q_8.

Question 10

5.2.iii
Show that Q_8 is not isomorphic to D_2n for every n >= 3.

Question 11

5.3
If m, n are coprime integers and G is an abelian group of order mn. Let H = {g^m : g \in G} and K = {g^n : g \in G}.

(ii) Show that any element of H has order dividing n, and any element of K has order dividing K. Hence, show that H \cap K = {id}.

(iii) Show that G = HK is isomorphic to H x K.

(iv) Show that #H = n and #K = m.

Question 12

5.4
Let m, n coprime. Show that C_m x C_n is isomorphic to C_mn.

Question 13

5.6.ii
If \phi: G -> H is an isomorphism, and N < H, K = \phi^{-1}(N). Show that G/K is isomorphic to a subgroup of H/N.

Question 14

7.1.ii
G non-abelian of order 12. Let a \in G have order 6. Let H = <a> and b \in G \ H. Show that bab^{-1} = a^{-1}></a>

Question 15

7.2.i
Let G act on X and x,y \in X, h \in G s.t. x = h*y. Show that Stab(x) = h Stab(y) h^{-1}.

7.2.ii
n >= 3. Find Stab_D_2n (i) for i \in {1, 2, …, n}.

Question 16

7.3
(H x K) x G _. G, (h, k) * g = hgk^{-1}. Deduce that #H #K = #(H \cap K) #HK.

Question 17

7.5.iii.b)
#G = p^2, #H = p, H x G/H -> G/H, (h, gH) -> h * gH = hgH. Show that Fix(H) = G/H.

Question 18

7.6.i
n>=2, 1<= k <= n. Show that the number of k-cycles in S_n of length k is n!/(k( (n-k)! ).

7.6.ii
Let o \in S_n be a k-cycle. Give formula for number of elements in S_n that commute with o.

Question 19

8.2
Let R be an I.D. Show that R[X]* = R*.

Question 20

8.6
Show that Z[\sqrt{-5}] is not a UFG.

Question 21

8.7
Show that the norm is a Euclidean function on Z[\sqrt{-2}].

Question 22

8.8
Let R be commutative ring, I = (a_1, …, a_m) and J = (b_1, …, b_n). Show that IJ = (a_{i}b_{j} : 1 <= i <= m, 1 <= j <= n).

Question 23

8.9
In R = Z[\sqrt{-5}], an ideal is either principal or has two generators. Verify that the following are in fact principal and find their generators.

i) I = (1 - \sqrt{-5}, \sqrt{-5})

ii) J = (2, 1+\sqrt{-5})^{2}.

Question 24

9.1
Divide 7 + 8i by 3 + 2i in Z[i].

Question 25

9.4
Show that I = (2, 1+\sqrt{-5}) is not a principal idael of Z[\sqrt{-5}]

Question 26

9.5
Let R = C[X_1, X_2, …], show that R is not Noetherian.

Question 27

9.6
Show that R is Noetherian iff. every ideal is finitely-generated.

Question 28

10.2
If M is an R-module, show that a finite subset X = {x_1, …, x_n} of M is an R-basis iff. every element of M can be written x = a_1 x_1 + … + a_n x_n with the a_i unique.

Question 29

10.3
n squarefree, G abelian of order n. Show that G is cyclic.

Question 30

10.4.i
If Z[i] free as a Z-module, what is its rank.

Question 31

10.4.ii
n >=2, is R^n free as a M_n(R)-module?

Question 32

10.5.ii
If A = (2 0 // 0 3) and B = (1 0 // 0 0), and M and N are R[X] modules where X acts as A in M, and X acts as B in N.
Let \phi:M -> N be a homomorphism or R-modules. Show that \pi = 0.

Question 33

10.6
Let M = Span_{Z}((6 // 4), (21 // 14)). Determine the structure of Z^{2} / M.