## Problem Sheet I: Defifinition and Examples of Groups

### 1.Consider the following two permutations from the symmetric group S7 of degree 7:  理科数学作业代写

Write each of the following permutations as products of disjoint cycles: (i) α; (ii) β; (iii) αβ;

(iv) βα; (v) α1; (vi) β1; (vii) (αβ)1; (viii) β1α1.

2.Show that the symmetric group Sn is non-abelian if and only if n > 3.  理科数学作业代写

3.Let F5 = {0, 1, 2, 3, 4} be the fifield of order 5, where addition and multiplication are calculated modulo 5. You may assume that F5 is a fifield.

Consider the general linear group GL2(F5), more often denoted by GL2(5).

(a) For , compute AB and A1.

(b) Prove that GL2(5) is non-abelian.

(c) Determine the order of GL2(5).

[Hint: A matrix is invertible if and only if its rows (or, equivalently, columns) are linearly  independent. How do you choose a linearly independent pair of vectors from F25?]

### 4.Let F be an arbitrary fifield. Show that the general linear group GLn(F ) is non-abelian if and only if n ≥2.  理科数学作业代写

5.Verify that the Klein 4-group is indeed a group.

[Do not check each of the 64 cases for associativity. Instead, use careful thought to reduce how many checks you need to perform.]

6.Consider the following three matrices with entries from the complex numbers:

(a) Let G = {I,-I,A,-A, B,-B,C,-C}, where I denotes the usual 2 × 2 identity matrix. Cal-culate a multiplication table for G and hence show that matrix multiplication defifines a binary operation on G.

(b) Deduce that the quaternion group is indeed a group.

### 7.Let G be a group and a be an arbitrary (but fifixed) element of G. Defifine a mapping τa: G →G by xτa= a−1xa for each x ∈G. (This mapping is called conjugation by a.)

(a) Specialise (for this part only) to the case G = S3 and a = (1 2 3). Compute στa for each σ ∈ S3.

(b) Prove that τa is always an isomorphism (in any group).

(c) If G is an arbitrary group, fifind an element a ∈ G such that τa is the identity mapping (that is, a = x for all x ∈ G).

(d) Prove that G is abelian if and only if all the mappings a are equal to the identity mapping.

8.Let G, H and K be groups.  理科数学作业代写

(a) If Ø: G → H is an isomorphism, show that its inverse Ø1 : H → G is also an isomorphism.

(b) If Ø: G → H and : H → K are isomorphisms, show that Ø : G → K is also an isomorphism.

(c) Show that ≅ (being isomorphic) is an equivalence relation on the class of all groups.

9.Let G be a group and defifine a mapping φ: G → G by = x1 for each x ∈ G.

(a) Prove that Ø2 (= ØØ) is the identity mapping.

(b) Prove that Ø is a bijection.

(c) Prove that Ø is an isomorphism if and only if G is abelian.

(d) Suppose that G is a fifinite group of even order. Prove that there exists a ∈ G of order 2.

### 10.Let G be any group. If x ∈G, prove that (x−1)−1= x.  理科数学作业代写

11.Let G be a group and suppose that x2 = 1 for all x∈G. Show that G is abelian.

12.Let G be a group and let a, b ∈ G. Prove that if a2= 1 and b2a = ab3, then b5 = 1.

[Hint: b4 = b2b2aa, b6 = b2b2b2.]

13.Let G be a group.

(a) Prove that for any x, y ∈G, the following formulae involving the order of an element hold:  理科数学作业代写

o(x) = o(x1); o(x) = o

(y1xy), and o(xy) = o(yx).

(b) If σ is a permutation in Sndecomposed as a product of disjoint cycles as σ = σ1σ2 … σk, where σi is a cycle of length ri, then prove that

o(σ) = lcm(r1, r2,…,rk).

(c) Consider the matrices

from the general linear group GL2(Q). Show that o(A) = 4, o(B) = 3, but that AB has infifinite order.