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MATH2022 LINEAR AND ABSTRACT ALGEBRA

创建日期：2024-04-11 15:55:20

所属分类：数学mathematics

标签： MATH2022

LINEAR AND ABSTRACT ALGEBRA

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MATH2022 LINEAR AND ABSTRACT ALGEBRA

This assignment comprises two questions and is worth 5% of the overall assess-
ment. It should be completed, scanned and uploaded using the MATH2022 Canvas
portal by 11:59 pm on Monday 24 May. Please do not include your name, as
anonymous marking will be implemented.
The first question establishes an important connection between two by two real ma-
trices and complex numbers, which is quite surprising if you have never seen it
before, and alleviates the stigma associated with the medieval “heresy” of taking
square roots of negative numbers.
The second question explores groups associated with nilpotent elements in modu-
lar arithmetic. Nilpotent elements play an important role in the theory of Jordan
canonical forms and more generally in advanced algebra.
Your tutor will give you feedback and allocate an overall letter grade (and mark)
using the following criteria:
A+(10): excellent and scholarly work, answering all parts of both questions, with clear
and accurate explanations and working, with appropriate acknowledgement
of sources, if appropriate, and at most minor or trivial errors or omissions;
A(9): excellent work, making progress on both questions, but with one or two sub-
stantial omissions, errors or misunderstandings overall;
B+(8): very good work, making progress on both questions, but with three or four
substantial omissions, errors or misunderstandings overall;
B(7): good work, making substantial progress on both questions, but making five or
six substantial omissions, errors or misunderstandings overall;
C+(6): reasonable attempt, making substantial progress on both questions, but mak-
ing seven or eight substantial omissions, errors or misunderstandings overall;
C(5): reasonable attempt, making progress on both questions, but making more than
eight substantial omissions, errors or misunderstandings overall;
D(4): making progress on just one question;
E(2): some attempt, but making no real progress on either question;
F(0): no real attempt at any question.
1. An isomorphism between groups H and K is a bijection ϕ : H → K that
preserves the group operations, that is, writing everything multiplicatively, and
writing the action of the bijection on the right,
(ab)ϕ = (aϕ)(bϕ)
for all a, b ∈ H. If this is the case, then we say that H and K are isomorphic,
and write
H ∼= K .
If H and K are groups then the Cartesian product of H and K is
H ×K = {(a, b) | a ∈ H , b ∈ K} ,
which becomes a group with coordinatewise group operations (and you do not
need to verify this). Throughout this exercise, put
G =
{[
a −b
b a
] ∣∣∣∣ a, b ∈ R , a2 + b2 6= 0} , C∗ = {z ∈ C | z 6= 0} ,
R+ =
{
a ∈ R ∣∣ a > 0} and C = {z ∈ C ∣∣ |z| = 1} .
(a) Verify that R+ is a group under multiplication. You may assume any of the
usual properties of real numbers.
(b) Verify that C is a group under multiplication. You may assume any of the
usual properties of complex numbers.
(c) Prove that G is an abelian group under matrix multiplication. You may
assume any of the usual properties of matrix arithmetic and determinants.
(d) Prove that, as multiplicative groups,
G ∼= R+ × C ∼= C∗ .
[Hint: use familiar properties of rotation matrices and polar forms of complex
numbers.]
(It follows that G and C∗ are isomorphic, because it is easy to check that
composites of isomorphisms are isomorphisms.)
2. Let n be a positive integer greater than 1.
An element x in Zn is called nilpotent if xk = 0 for some positive integer k.
For example, 0 and 2 are nilpotent in Z4, and 0, 3 and 6 are nilpotent in Z9.
Let N = N(Zn) denote the set of all nilpotent elements in Zn and put
1 +N = {1 + z | z ∈ N} .
(a) Prove that N is closed under addition and multiplication.
(b) Verify that N is an abelian group with respect to addition.
(c) Prove that N is a group with respect to multiplication if and only if n is
not divisible by the square of a prime number, in which case the group N is
trivial.
(d) Prove that 1 +N is a group with respect to multiplication.
(e) Verify that 1+N(Z27) is a cyclic group under multiplication, but 1+N(Z16)
is not a cyclic group under multiplication.

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