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MATH3907 ⋅ SESSION 1, 2022 ASSIGNMENT 1 ⋅ SOLUTIONS
发布时间:2022-07-09
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MATH3907 ⋅ SESSION 1, 2022
ASSIGNMENT 1 ⋅ SOLUTIONS
1. [6 marks]. Consider on the set ℤ × ℤ the operation ∗ defined by
( +
, ℓ −
) if
is odd.
Show that (ℤ × ℤ, ∗) is a group. You may assume that ∗ is an associative operation.
(![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps4.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps5.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps6.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps7.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps8.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps9.jpg)
![](file://C:/Users/ADMINI~1/AppData/Local/Temp/ksohtml89076/wps10.jpg)
2.Let
be a group, and let
∶
→
be the function with
(
) =
−1.
(a) [3 marks]. Show that if is abelian, then the function
is a group isomorphism. (b) [2 marks]. Show that, conversely, if
is a group isomorphism then
is abelian.
3. Consider the elements ,
∈
5 given in two-line notation by
|
3 5 |
4 3 |
|
|
3 1 |
4 3 |
|
(a) [2 marks]. Write and
in cycle notation, and compute their product. (Remember that for us,
∗
means“first do
, then
”)
(b) [1 marks]. Compute the conjugate permutation =
.
(c) [2 marks]. Find another permutation , different from
, such that
=
.
(d)∗ [1 mark]. In total, how many permutations are there which, like and
, conjugate
into
?
4.(a) [2 marks]. Explain why a permutation is even if and only if it contains an even number of even- length cycles.
(b) [2 marks]. List all the even cycle-types (apart from the identity) in 6.
(c) [4 marks]. Calculate the number of elements of each of these cycle-types. Show your working, and briefly explain how you get your answers. (Make sure that, together with the identity, they sum to the order of 6, which is 360.)
(d)∗ [2 marks]. Give two different explanations of the following fact:
For any permutations ,
∈
, the parity of
is the same as the parity of the conjugate
.
5.Let
,
∈
2(ℂ) be the complex-valued 2 × 2 matrices given by:
= ( )
= ( )
where is the complex number
/4.
(a) [3 marks]. Show that has order 2 and
has order 8 in
2(ℂ).
(b) [2 marks]. Verify the equation =
3
.
(c) [3 marks]. Use your answers to (a) and (b) to explain why the subgroup of 2(ℂ) generated by
and
is the group of order 16 with elements
{ ,
,
2, … ,
7,
,
,
2
, …
7
} .