## Description

ve203 Assignment 5

Q1.

(i)* How many different anagrams can be made from the word “unnecessarily”.

(ii) Prove that the number of ways of distributing n distinguishable objects into k

boxes, A1, . . . , Ak, such that for all 1 ≤ i ≤ k, ni objects appear in box Ai

is

n!

n1! · · · nk!

(iii) How many surjections are there from [5] to [3]?

(6 marks)

Q2. Let (G, ·) be a group.

(i) Prove that the identity element of (G, ·) is unique.

(ii) Prove that for all x ∈ G, x has a unique inverse.

(2 marks)

Q3. Write the following bijections as products of disjoint cycles and state their order

in the group S9:

(i) f : [9] −→ [9] defined by: for all n ∈ [9],

f(n) =

n + 3 if n + 3 < 9,

n + 3 − 9 if n + 3 ≥ 9

(ii) (13)(203)(16)(38)(14)(234)

(iii) (1203)(245)(231)(105)

(iv) (45)(123)(456)(12)

(4 marks)

Q4. Write the following bijections as products of 2-cycles and state whether they are

even or odd:

(i) (1256)(12439)

(ii) f : [9] −→ [9] defined by: for all n ∈ [9],

f(n) =

n + 2 if n + 2 < 9,

n + 2 − 9 if n + 2 ≥ 9

(iii) (0124)(2198)(132568)

(iv) (120)(94567)(0427)

(4 marks)

1

Q5. For the following sets G and binary operations ? : G × G −→ G either prove that

(G, ?) is a group or show that (G, ?) is not a group:

(i) G = {x ∈ R | x > 0} and x ? y =

√xy

(ii) G = R\{0} and x ? y =

x

y

(iii) G is the set of all 2 × 2 matrices and ? is matrix multiplication

(iv) G = {x ∈ Q | x > 0} and x ? y =

xy

2

(4 marks)

Q6. Let n ≥ 3 and consider Sn.

(i) We say that a 2-cycle (pq) is adjacent if p = k and q = k + 1. Prove that for all

σ ∈ Sn, if σ can be written as an odd number of 2-cycles, then σ can be written

as an odd number of adjacent 2-cycles, and if σ can be written as a product of

an even number of 2-cycles, then σ can be written as an even number of adjacent

2-cycles.

(ii) For all σ ∈ Sn, define

P(σ) = |{(k, l) ∈ [n] × [n] | (k < l) ∧ (σ(l) < σ(k))}|

Prove that if (pq) is an adjacent cycle and σ ∈ Sn, then P((pq)σ) = P(σ) ± 1.

(iii) Use (ii) to prove that no σ ∈ Sn is both even and odd.

(iv) The Alternating Group on [n], denoted An, is the set of all even bijections in Sn.

Prove that An is a subgroup of Sn.

(v) Prove that |An| =

n!

2

.

(12 marks)

Q7. Let (G, ·) be a group. Let x, y ∈ G be such that xyx−1 = y

2 and y 6= e.

(i) Show that x

5yx−5 = y

32

.

(ii) If the order of x is 5, then what is the order of y? Justify your answer.

(6 marks)

Q8. Find a group (G, ·), x, y ∈ G and n ∈ N\{0, 1} such that

(xy)

n

6= x

n

y

n

(2 marks)

Q9. Let (G, ·) be a group. Prove that if for all x ∈ G, x

2 = e, then (G, ·) is abelian.

(2 marks)

Q10. Find all of the subgroups of D4.

(4 marks)

2