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ve203 Assignment 5

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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)

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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