Download Problem Sheet 4

Problem Sheet 4
1. F Let
σ1 =
σ2 =
σ3 =
τ1 =
τ2 =
1 2 3 4 5 6
2 4 5 1 3 6
1 2 3 4 5 6
1 4 3 6 2 5
1 2 3 4 5 6
6 3 2 1 4 5
∈ S6 ,
∈ S6 ,
∈ S6 ,
1 2 3 4 5 6 7 8 9
7 1 9 8 6 4 2 5 3
1 2 3 4 5 6 7 8 9
1 7 5 9 8 4 2 3 6
∈ S9,
∈ S9 .
a) Calculate σ 1 σ 2 , σ 2 σ 3 , σ 3 σ 1 , σ 21 , σ 33 , σ 1 σ 2 σ 1 , τ 1 τ 2 , τ 2 τ 1 τ 2 and
τ 41 .
b) Find the inverses of σ 1 , σ 2 , σ 1 σ 2 , σ 3 , τ 1 and τ 2 .
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c) Verify that (σ 1 σ 2 )−1 = σ −1
2 σ1 .
2. a) Find subsets of S3 closed under composition. (That is, find sets
S ⊆ S3 such that if σ, τ ∈ S then σ ◦ τ and τ ◦ σ ∈ S.)
b) Construct the composition tables for
(i) 1 2 3 4
1 2 3 4
1 2 3 4
14 ,
,
,
⊆ S4 ,
2 1 4 3
3 4 1 2
4 3 2 1
(ii) 14 ,
1 2 3 4
2 4 1 3
1 2 3 4
1 2 3 4
,
,
⊆ S4 .
3 1 4 2
4 3 2 1
c) Can you find a subset of S4 , containing six elements, which is closed
under composition?
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3. Count the number of elements in the following subsets of S5 .
(i) The set of transpositions.
(ii) The set of cycles of length 3.
(iii) The set of permutations that fix a given element?
Which of the above subsets are closed under composition?
4. Write the cycle (1, 2, 3, 4) ∈ S4 as a product of transpositions in two
different ways.
5. F Calculate the orders of the following permutations in S5 :
(i) (1, 2, 3) (1, 3, 4) (1, 3, 5) ,
(ii) (1, 2) (1, 3) (1, 4) (1, 5) ,
(iii) (2, 3, 5) (1, 2) (2, 4) (1, 2) .
6. a) Write the five permutations in Question 1 as products of distinct
cycles.
b) Calculate the orders of these permutations.
7. Give an example of a permutation in S11 with the largest possible order.
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8. F Use the table below to sieve the integers up to 200 for primes. Thus
calculate π (200) .
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
2
12
22
32
42
52
62
72
82
92
102
112
122
132
142
152
162
172
182
192
3
13
23
33
43
53
63
73
83
93
103
113
123
133
143
153
163
173
183
193
4
14
24
34
44
54
64
74
84
94
104
114
124
134
144
154
164
174
184
194
5
15
25
35
45
55
65
75
85
95
105
115
125
135
145
155
165
175
185
195
6
16
26
36
46
56
66
76
86
96
106
116
126
136
146
156
166
176
186
196
7
17
27
37
47
57
67
77
87
97
107
117
127
137
147
157
167
177
187
197
8
18
28
38
48
58
68
78
88
98
108
118
128
138
148
158
168
178
188
198
9
19
29
39
49
59
69
79
89
99
109
119
129
139
149
159
169
179
189
199
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
What is the smallest composite number that has no prime factor in the
table?
Are the following numbers prime or composite?
(i) 44517
(ii) 44503
(iii) 44519.
9. Show that if n ≥ 2 then the numbers
n! + 2, n! + 3, ..., n! + n
are all composite.
This shows that are gaps as long as you would wish for in the sequence
of primes.
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10. (i) Find the smallest four sets of prime triplets of the form p, p + 4 and
p + 6.
(ii) Why are there no prime triplets of the form p, p + 2 and p + 4, other
than (3, 5, 7)?
(Hint, look at the p modulo 3.)
11. Calculate Euler’s phi function, φ (n) , for each 1 ≤ n ≤ 12.
What is φ (p) for prime p?
What is φ pk for k ≥ 1 and prime p?
12. F Use Fermat’s Little Theorem and
(i) show that 55552222 + 22225555 is divisible by 7,
(ii) show that 55552222 + 22225555 is divisible by 3 but not by 9,
(iii) find the last two digits in the decimal expansion of 33337777 +
77773333 .
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