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MAT301 Week 2 Exercises

If G is a group, we will denote the order of g ∈ G by o(g) or ord(g).

Exercise 1. Using one of the subgroup tests, prove that H = 0(1) 1(a) : a ∈ Z is a subgroup of SL2 (Z).

Exercise 2. Using one of the subgroup tests, prove that

H = : x,y,z ∈ R

is a subgroup of SL3 (R). The group H is called the Heisenberg group.

Exercise 3. Let p be a prime number and let Fp = Z/pZ. Consider GLn (Fp ), the set of all invertible n × n matrices with entries in Fp . What is the order of GL2 (Fp )?

Exercise 4 (Intersections of subgroups are subgroups). Let G be a group.

1. Let H1 and H2 be subgroups of G. Prove that H1 ∩ H2 is a subgroup of G.

2. Let {Hi }i∈I be an arbitrary set of subgroups of G. Prove that Ti∈I Hi is a subgroup of G.

Exercise 5 (Unions of subgroups). Let G be a group and let H1 ,H2 ≤ G. Prove that H1 ∪ H2 ≤ G if and only if H1 ⊆ H2 or H2 ⊆ H1 .

Exercise 6. The Pauli matrices are the matrices

σ 1 = 1(0) 0(1) , σ2 = , σ3 =

in GL2 (C). The Pauli group G1 is defined to be the subgroup of GL2 (C) generated by σ 1 ,σ2 ,σ3 , that is, G1 = ⟨σ1 ,σ2 ,σ3 ⟩ . List all of the elements of G1 .

Exercise 7. Let S = and let T = 0(1) 1(1) . We will prove that SL2 (Z) = ⟨S,T⟩ .

Let A = c(a) d(b) ∈ SL2 (Z).

1. Prove that every upper triangular matrix in SL2 (Z) is of the form ±Tn for some n ∈ Z.

2. Compute SA and Tn A for all n ∈ Z.

3. Prove that there exists n ∈ Z such that STn A = c′(c) d′(d) ∈ SL2 (Z) for some c′ ,d′ ∈ Z with 0 ≤ c′ < |c| .

4. Prove that there exists a finite sequence n1 , . . . ,nr ∈ Z such that STnr . . . STn1 A is an upper triangular matrix in SL2 (Z). Conclude that A ∈ ⟨S,T⟩ .

Exercise 8 (Dicyclic groups). Let n ≥ 2 be an integer. The dicyclic group Dicn is the subgroup of the multiplicative group of quaternions H × generated by ζ = eiπ/n and j . That is, Dicn = ⟨ζ,j⟩ . Note that Dic2 = Q8 , the quaternion group. Some people call Dicn a generalised quaternion group and denote it by Q4n . Some people only call Dicn a generalised quaternion group if n is a power of 2.

1. Prove that ζ 2n = 1, j2 = ζ n , and jζj−1 = ζ −1 .

2. Prove that every element of Dic2n can be written uniquely in the form ζ kjℓ with k ∈ {0, . . . , 2n − 1} and ℓ ∈ {0, 1} . Conclude that |Dicn | = 4n.

3. Compute the centralizer of every element of Dicn . What is the centre of Dicn ?

Exercise 9 (centralizers). Let G be a group. For each subset S ⊆ G, we define the centralizer of S in G to

be the subset CG (S) of G consisting of all elements of G that commute with all elements of S. That is, CG (S) = {g ∈ G | ∀ s ∈ S,gs = sg} .

Prove that CG (S) is a subgroup of G. If S = {s} ⊆ G, then CG (S) is denoted by CG (s) and called the centralizer of s in G, which is denoted by C(s) in your textbook. Also, note that the centralizer of G in G is the centre of G, that is, CG (G) = Z(G).

Exercise 10 (Gauss’s formula for divisor sums of φ). Let G be a cyclic group of order n. For each d ∈ Z>0 , let Sd denote the set of elements of G of order d. Then we have a disjoint union G = d∈Z>0 Sd . Use this to prove that n = d|n φ(d).

Exercise 11 (Number of Elements of order d in a finite Group). Let G be a finite group and let d ∈ Z>0 . Define cd (G) to be the number of cyclic subgroups of G of order d. Prove that the number of elements of

order d in G is cd (G)φ(d).

Exercise 12. Let G be a group.

1. Prove that for all a ∈ G we have o(a) = o(a−1).

2. Prove that for all a,b ∈ G we have o(ab) = o(ba) and o(bab−1) = o(a).

Exercise 13. Let G be a group and suppose that there is a unique element a ∈ G of order two. Prove that a ∈ Z(G).

Exercise 14 (The order of a product of commuting elements). Let a,b ∈ G and assume that ab = ba.

1. Assume that o(a),o(b) < ∞ . Prove that o(ab) < ∞ and o(ab) divides lcm(o(a),o(b)).

2. Assume that o(a),o(b) < ∞ . Prove that divides o(ab) and that |⟨a⟩ ∩ ⟨b⟩| divides gcd(o(a),o(b)). It follows that divides , which in turn divides o(ab).

To summarise, if o(a),o(b) < ∞, then o(ab) < ∞ and

lcm(o(a),o(b)) lcm(o(a),o(b))

gcd(o(a),o(b)) |⟨a⟩ ∩ ⟨b⟩|

3. Prove that if o(a),o(b) < ∞ and gcd(o(a),o(b)) = 1, then o(ab) = o(a)o(b).

4. Prove that if o(a) < ∞ and o(b) = ∞, then o(ab) = ∞ .

5. Assume that o(a) = ∞ . Give an example of an element b ∈ G such that ab = ba and o(b) = ∞, but o(ab) < ∞ .

Exercise 15. Let n ∈ Z ≥1 ∪ {∞} . Give an example of a group G and elements a,b ∈ G such that o(a) = o(b) = 2 and o(ab) = n.

In fact, the following is true.

Theorem 1. Let i,j ∈ Z>1 ∪ {∞} and k ∈ Z≥1 ∪ {∞} . There exists a group G and elements a,b ∈ G such that o(a) = i, o(b) = j, and o(ab) = k . If i,j,k are all finite, then the group G can be taken to be finite .

Exercise 16 (The exponent of a group and a characterisation of cyclic groups) . Let G be a group and assume that {o(a) : a ∈ G} is finite (which is true if G is finite). The exponent of G is defined by exp(G) = lcm{o(a) : a ∈ G} .

1. Prove that exp(G) is the smallest k ∈ Z>0 such that ak = e for all a ∈ G.

2. Assume that G is cyclic. Prove that exp(G) = |G| .

3. Assume that G is abelian. Prove that exp(G) = max{o(a) : a ∈ G} . Conclude that there exists a ∈ G with o(a) = exp(G).

4. Assume that G is abelian. Conclude that G is cyclic if and only if exp(G) = |G| .