Exercise 1

Determine in GL₂(R)

• a matrix A mapping the vector (1,0)^T onto (0,1)^T,
• a matrix B mapping (0,1)^T onto (1,1)^T, and
• a matrix C mapping (1,0)^T onto (1,1)^T.

Exercise 2

Let G be a group and g in G. In analogy with the map L_g (left multiplication by g), a map R'_g : G -> G can be defined:

R'_g(h) = hg      (h in G).

1. Prove that for each g in G the map R'_g is in Sym(G).
2. Does the map from G to Sym(G), given by g -> R'_g, define a morphism?

Exercise 3

Suppose G is a subgroup of Sym(X) for some set X. Then being in an orbit of G defines an equivalence relation. This does not hold when G is a monoid and not a group as will be clear from the following.

Let X be the set Z and M the monoid (N,+,0). Define

f : M -> Sym(Z)

by

(f(n))(k) = k + 3n,     (n in N, k in Z).

1. Show that f is a morphism of monoids.
2. Define the relation ~ on Z by x ~ y if and only if there exists an n in N with (f(n))(x) = y. Show that this relation is not symmetric.

Exercise 4

The group O_n(R) of orthogonal matrices acts on Rⁿ by left multiplication. Show that an orbit consists of all vectors of Rⁿ with a fixed length. So there infinitely many orbits.

Exercise 5

Matrices in GL₂(R) transform lines through the origin in R² into lines through the origin. Determine the stabilizer H of the x-axis. Determine also the stabilizer in H of the y-axis. What is the kernel of the action on the lines?

Exercise 6

Consider the permutation representation of S₃ on the left cosets in S₃/<(1,2,3)>. What is the kernel and what is the image of this permutation representation?

Exercise 7

Determine a basis for the automorphism group of the square. Determine the order of the automorphism group of a square. Describe also the action of this group on the two diagonals of the square.

Exercise 8

Prove that

<(1,2),(1,2,3,4)> = <(2,3),(1,2,3,4)> = <(3,4),(1,2,3,4)> = <(1,4),(1,2,3,4)> = S₄.

What is the order of the subgroup H = <(2,4),(1,2,3,4)> of S₄? Provide an isomorphism from the group H to the group D₄ of automorphisms of the square.

Exercise 9

Show that the group D_n of symmetries of a regular n-gon contains n rotations and n reflections. Determine a basis for D_n. What is the order of D_n?

Exercise 10

Let G be the automorphism group of the tetrahedron. Determine a basis for G and use it to find the order of G.
Same question for the cube. Describe also the action of the automorphisms on the 4 diagonals of the cube.

Exercise 11

Let a be the positive real fourth root of 2 so that a is a root of X⁴ - 2. Determine all automorphisms of Q(a).

Exercise 12

Prove the following equivalence for a subgroup N of G:

gN = Ng for all g in G     iff     g^-1ng in N for all g in G and n in N.

Exercise 13

Let f: G -> H be a morphism of groups. Show, by means of an example, that the image f(G) need not be a normal subgroup of H.

Exercise 14

The subgroup K = <(1,2)(3,4),(1,3)(2,4)> of S₄ is called Klein's Vierergroup.

1. Establish that K has order 4 and is a normal subgroup of S₄ as well as A₄.
2. Verify that K is isomorphic to Z/2Z × Z/2Z.
3. Give a non-normal subgroup of S₄ that is also isomorphic to Z/2Z × Z/2Z.

Exercise 15

Let H be a subgroup of the group G.

1. Show that each normal subgroup N of G contained in H is also contained in the kernel of the morphism L : G -> Sym(G/H).
2. Show that H is a normal subgroup if G has order 9.

Exercise 16

Let G be a group.

1. Determine the conjugacy class of the unit element e.
2. In this part, G = S₄. Determine the conjugacy classes of each of the following elements: (1,2,3), (1,2,3,4), (1,2)(3,4).
3. Show that each conjugacy class consists of a single element if G is commutative. Prove also the converse: if each conjugacy class consists of exactly one element, then G is commutative.
4. Prove that all elements from the same conjugacy class have the same order.

Exercise 17

The center, denoted by Z(G), of a group G is the set

Z(G) = {g G | hgh^-1 = g for all h G}.

1. Show that Z(G) is a normal subgroup of G.
2. Determine the center of the group S₃.
3. What is the center of a commutative group?
4. Determine the conjugacy class of an element from the center.
5. If a is an element of G of order 2 and <a> is a normal subgroup of G, then a is an element of Z(G).
6. Show that scalar multiplication by a nonzero a is contained in the center of GL₂(R).
7. Prove that the center of GL₂(R) consists of scalar matrices only.

Exercise 18

Consider the quotient group Q/Z.

1. Show that each element of the group has finite order.
2. Establish that the group itself has infinite order.
3. What is the order of the element 28/16 + Z?
4. What is the order of an arbitrary element a/b + Z?

Exercise 19

The quotient group S₄/K, where K = <(1,2)(3,4),(1,3)(2,4)> is Klein's Vierergroup, is isomorphic with a group of order 6. Which one? S₃ or Z/6Z?

Exercise 20

Let C^* be the multiplicative group of the complex numbers distinct from 0.

1. Show that H = {z C | |z| = 1} is a subgroup of C^*.
2. Show that the map f: R -> H,   f(t) = e^2it is a surjective morphism.
3. Prove that R/Z is isomorphic to H.

Exercise 21

Use the table of groups of order at most 10 as given in Section 8.6 when answering the following questions.

1. Which groups from the table are commutative?
2. Let G be a group of order 8 generated by two elements a and b of order 2 with (ab)⁴ = e. With which group of order 8 from the table is G isomorphic?
3. Which groups of order 8 are (isomorphic to) subgroups of S₄?

Exercise 22

Determine all groups of order 15 up to isomorphism.

Exercise 23

Let G be the group Z/2Z × Z/2Z. Call its nontrivial elements a, b, c and, as usual, let e be the unit element.

1. Describe the left regular representation of G.
2. Describe the action by left multiplication on the set X₂ of subsets of G consisting of two elements. Is the action transitive?

Exercise 24

Let G be a group.

1. Show that, for each g G, the map
   C'_g: G -> G,       C'_g(x) = g^-1xg

   is in Sym(G).

2. Is the map
   C' : G -> Sym(G),       g -> C'_g

   a permutation representation?

3. Can you make a permutation representation with
   M_g : G -> G,       M_g(x) = gxg?

Exercise 25

Consider S_n and let X be the set of all subsets of {1, ..., n}. There is an obvious permutation representation f : G -> Sym(X), defined by g({a₁ , ..., a_m})= { g(a₁), ..., g(a_m)}, where {a₁, ..., a_m} is a subset of X. Determine the orbits of S_n. Do the same for A_n acting on X. (Watch out for n=2.)

Exercise 26

Define a map S₂ -> Sym(R²) by g (x₁ ,x₂) = (x_g^-1(1), x_g^-1(2)) for g in S₂.

1. Show that f is a permutation representation.
2. What are the orbits of S₂ on R²?
3. For which vectors in R² is the stabilizer equal to S₂?
4. What is the geometric significance of the action of (1,2)?

Exercise 27

Report on similar questions as in Exercise 26 for the action of S₃ on R³:

h : S₃ -> Sym(R³)     with     (h(g))(x₁, x₂, x₃) = (x_g^-1(1), x_g^-1(2), x_g^-1(3)) for g in S₃. Determine also the vertices for which the stabilizer is equal to <(1,2)>. Which permutations act as a rotation on R³ (determine the axis and angle of rotation)? Which act as reflections?

Exercise 28

Let X = {(x₁ ,x₂ ,x₃,x₄) R⁴ | x₁ +x₂ +x₃ +x₄ = 0}. Define a permutation representation of S₄ on X by setting, for g S₄,

g(x₁, x₂, x₃, x₄) = (x_g^-1(1),x_g^-1(2),x_g^-1(3),x_g^-1(4)).

1. Prove that this indeed a permutation representation.
2. Show that each g S₄ acts as a linear transformation on X and even as an orthogonal transformation.
3. Show that (1,2) acts as a reflection and (1,2,3,4) as a product of a reflection and a rotation.

Exercise 29

Consider G = Z × Z. Define f : G -> Sym(C) by f(m,n)(z) = z· i^m+n.

1. Show that f is a permutation representation.
2. Determine the kernel of f.
3. Determine the orbits of g on C.
4. Same questions as in part 1, but with i replaced by e^2i/5.

Exercise 30

Laat V = {(x₁, ..., x₆) in (Z/2Z)⁶ | x₁+ ··· +x₆ = 0}. Define a permutation representation f : S₆ -> Sym(V) by

g(x₁, ... , x₆) = (x_g^-1(1), ... ,x_g^-1(6)).

1. Show that this is indeed a permutation representation.
2. Show that the stabilizer of (1,1,0,0,0,0) is isomorphic to S₂ × S₄.
3. Determine the orbit of (1,1,0,0,0,0).
4. Generalise to the case where the group S_n (with n even) acts on the set W = {(x₁, ... , x_n) (Z/2Z)ⁿ | x₁ + ··· + x_n = 0}. Deduce from this, by studying the orbits, that 2^n-1 = 1 + (n!/(2! (n-2)!)) + (n!/(4!(n-4)!)) + ··· + (n!/((n-2)!2!)) + 1.

Exercise 31

1. Let S¹ = {z C | |z|=1}, the circle of radius 1 around 0 in C. Define a map f : Z -> Sym(S¹) by (f (n))(z) = iⁿz (in short, n(z) = iⁿz). Show that f is a permutation representation and determine its kernel.
2. Show that the vertices on S¹ of a square form an invariant set.
3. Describe the action of the subgroup 2Z of Z on S¹. What are the invariant subsets of S¹ under the action of 2Z?

Exercise 32

Let G = GL₃(R) and

X ={ {u, v} | u and v are independent vectors in R³}.

1. Show that, for each A GL₃(R), the map X -> X, {u, v} -> {Au, Av} is a bijection.
2. Define a permutation representation of G on X as suggested by the previous part. Is it injective? Is it transitive?
3. Same questions as before for X consisting of triples of independent vectors in R³.

Exercise 33

1. Let G = S₃ and H = <(1,2)>. Describe the left regular representation L : G -> Sym(G/H).
2. Same question for G = S₄ and H a subgroup of order 4. (There are two different subgroups of order 4!)

Exercise 34

Show that a group of order n cannot act transitively on a set with more than n elements.

Exercise 35

Consider the automorphism group G of a regular octahedron.

1. Show that the automorphism group acts transitively on the set of vertices.
2. Show that the stabilizer of each vertex has order 8. What is the connection with the automorphisms of a square? What is the order of G?
3. Describe the action of G on the three diagonals of the octahedron. Is the morphism G -> Sym(D), where D is the set of diagonals, surjective?
4. Is the action of G on the centers of gravity of the 8 faces of the octahedron an injective permutation representation? Do you spot a connection with the cube?
5. Does G act transitively on the set of all unordered pairs of vertices?

Exercise 36

Consider a transitive permutation group G on a set X. Show that, for each x and x' from X, the stabilizers G_x and G_x' are conjugate, that is, there is g G with gG_xg^-1 = G_x'.

Exercise 37

In this exercise we determine the automorphisms of the field Q(a), where a = i + 2.

1. Show that a² - 2i a = 3. Deduce from this that i Q(a).
2. Prove that 2 also belongs to Q(a).
3. Conclude that Q(a) = Q(i, 2).
4. Determine a polynomial f Q[X] of degree 4 having a as a root.
5. What are the zeros of f in C?
6. Determine all automorphisms of Q(a); describe such an automorphism by its image on a.
7. Construct the multiplication table of this group. Is it a cyclic group? Indicate the images of i and 2 under each automorphism.

Exercise 38

Let z = e^{2 i/5}.

1. Show that z is a root of the polynomial X⁵ - 1 in Q[X]. What are the roots of this polynomial in C?
2. Determine a polynomial f Q[X] of degree 4 having root z.
3. Determine the automorphism group of Q(z) and show that this group is cyclic.

Exercise 39

In which of the following cases is the group N a normal subgroup of the group G?

1. N =< (2,3)>, G = S₄.
2. N =< (1,2,3,4)>, G = S₄.
3. N is the subgroup of all rotations in the automorphism group G of a regular 5-gon.

Exercise 40

Determine all normal subgroups of S₃.

Exercise 41

Put G = GL₂(R).

1. Show that the diagonal matrices D form a subgroup which is not a normal subgroup of G.
2. Prove that the diagonal matrices of the form aI with nonzero a do form a normal subgroup of G.
3. Is the set of upper triangular matrices a normal subgroup of G?

Exercise 42

Let G be a group.

1. Prove: if N and M are normal subgroups of G, then so is N M.
2. Prove: if N is a normal subgroup of G and H a subgroup of G, then N H is a normal subgroup of H.
3. Show, by means of groups G = S₄, N = A₄ and a suitable subgroup H, that N H need not be a normal subgroup of N.
4. Show, by means of an example, that the following assertion does not hold in general:

   If N is a normal subgroup of H and H a normal subgroup of G, then N is a normal subgroup of G.

5. Show: If H is a subgroup of G and g G, then g^-1Hg is also a subgroup of G.
6. If, moreover, H is the only subgroup of G of order n, then H is a normal subgroup of G.

Exercise 43

Let G be a finite group, generated by the set B and suppose H is a subgroup of G generated by A. Show that H is a normal subgroup of G if and only if b^-1ab in H for all b B and all a A.

Exercise 44

Suppose f :G -> H is a morphism of groups.

1. Prove: if N is a normal subgroup of H, then f^-1(N) is a normal subgroup of G.
2. If f is surjective and N is normal in G, then f(N) is normal in H. Show, by means of an example, that the surjectivity condition cannot be removed.

Exercise 45

If G is a group and H a subgroup of G of index 2, then H is normal in G. Prove this in each of the following two ways:

1. By comparing left cosets and right cosets of H in G,
2. By use of the left regular representation G -> Sym(G/H).

Establish also that, for each g, h G the intersection {g, h, gh} H is not empty.

Exercise 46

Let G be a group of order 2p where p is an odd prime number.

1. Show that G contains a normal subgroup H of order p.
2. Prove that G contains an element, g say, of order 2.
3. If h H is not 1, and gh = hg, then gh is an element of order 2p. Give a proof of this assertion and conclude that in this case G is cyclic and hence isomorphic to Z/2pZ.
4. Let h H. Prove: If gh hg, then gh' h'g for all h' H with h' 1.
5. Show that for all h H, the element ghg belongs to H; derive from this that (ghg)h = h(ghg).
6. Let f = ghgh. Prove that gf = fg.
7. Verify that, for all h H: If gh hg, then ghg = h^-1.
8. Show that G is isomorphic to the automorphism group of a regular p-gon if G is not cyclic.

Exercise 47

Let G be a group and X a subset of G. The normaliser N_G(X) of X in G is the set of elements g of G with

gXg^-1 = X.

Notice that N_G(X) is a subgroup of G. Show that <X> is a normal subgroup of N_G(X).

Exercise 48

Determine all normal subgroups of S₄.

Exercise 49

Suppose G is a group. Let g be an element of G. The centraliser C_g(G) of g is the set of elements in G commuting with g, that is,

C_g(G) = {h G | hg = gh}.

1. Show that C_g(G) is a subgroup of G containing <g>.
2. What is the centraliser of g if G is commutative?
3. When do we have |C_G(g)| = 1?
4. Compute the centraliser of (1,2) in S₄.
5. Prove that the number of elements in the conjugacy class of g is equal to |G|/|C_G(g)|. Conclude that this number is a divisor of |G|.

Exercise 50

Prove in each of the following cases that N is a normal subgroup of G, and that H is isomorphic to G/N.

1. G = C^*, N = {z C | |z| = 1} and H = R⁺ with the operation multiplication.
2. G = R^*, N = {-1,1}, and H = R⁺ with the operation multiplication.
3. G = C^*, N = R⁺, and H = {z C | |z| = 1}.
4. G = Z × Z, N = mZ × nZ, and H = (Z/mZ) × (Z/nZ).
5. G = Q, the quaternion group, N = {1,-1}, and H = Z/2Z × Z/2Z.
6. G is the set of all invertible 2 × 2 matrices with entries from Z/7Z; N is the subgroup of those matrices having determinant in {1,-1}, and H = Z/3Z.

Exercise 51

Let G be a permutation group on the set X. The group G is called t-transitive, where t in N, if it is transitive on the ordered t-tuples from X.

1. Prove that G is t-transitive if and only if the stabilizer of each s-tuple of elements from X (where s < t) is transitive on the remaining elements of X.
2. Show that S_n is n-transitive and that A_n is (n-2)-transitive on {1, ... , n}.

Exercise 52

Suppose that G is a 2-transitive permutation group on {1, ... , n} with n > 1.

1. Show that G = S_n if G contains a transposition.
2. Show that G = A_n or S_n if G contains a 3-cycle.

Exercise 53

GL_n(R), n > 2, can be viewed as a permutation group on the set X of 1-dimensional subspaces of Rⁿ. If g G and x X, then g(X) = {g(v) | v X}.

1. What is the kernel of this permutation representation?
2. Show that if n = 2, the group G acts 3-transitively on X.
3. For n > 2, the group G is 2-transitive but not 3-transitive. Prove this.

Exercise 54

Label the vertices as in the following figure and consider the game in which you are allowed to rotate each of the 4 small triangles.

Prove that these moves generate the subgroup A₆ of S₆.

Exercise 55

Label the vertices of a 2 by 2 by 2 cube with the integers 1, 2, 3, 4, 5, 6, 7, 8 as shown in the figure.

Consider the following game: each single move consists of turning a face of the cube over 90 degrees (clockwise or counter clockwise). How many different positions can be obtained by applying such moves?