THE UNIVERSITY OF SYDNEY
MATH2022 LINEAR AND ABSTRACT ALGEBRA
Semester 1 First Assignment 2021
This assignment comprises two questions and is worth 5% of the overall assessment.
It should be completed, scanned and uploaded to Canvas by 11:59 pm on Friday 23
April. Please do not include your name, as anonymous marking will be
implemented.
The first question explores properties of idempotents in modular arithmetic and
related subgroups.
The second question is an application of the conjugation principle to solve an ap-
parently difficult problem in planar geometry.
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. In this exercise, we explore idempotents in Zn and some associated subgroups.
If G is a group with respect to the binary operation ∗ then we call g ∈ G an
idempotent if g ∗ g = g.
If n is a positive integer then we call z ∈ Zn an idempotent if z2 = z (which is
analogous to the definition in the previous sentence, but the binary operation
being used in the context of Zn is always multiplication).
We call u ∈ Zn a unit if uv = 1 for some v ∈ Zn.
Throughout this exercise, you may use the fact (without proof) that multiplica-
tion in Zn is associative and commutative.
(a) Prove that any group contains exactly one idempotent.
(b) Find all of the idempotents in Z6, Z10, Z15 and Z30. To get full credit it is
enough to list them correctly (and you do not need to justify your answers).
(c) Verify that the set U of units in Zn is closed under multiplication, so that
multiplication becomes a binary operation on U . Now verify that U is an
abelian group under multiplication.
(d) Let e = e2 be an idempotent in Zn. Verify that
eU = {eu | u ∈ U}
is an abelian group with respect to multiplication, where U is the set of units
in Zn.
(e) Find two disjoint subsets G1 and G2 of Z30, each with 8 elements, such that
both G1 and G2 are abelian groups with respect to multiplication.
(f) Show that the groups G1 and G2 that you found in the previous part are not
cyclic.
2. Fix x0, y0, θ ∈ R and define bijections T,R : R2 → R2 by the rules
T (x, y) = (x− x0, y − y0)
and
R(x, y) = (x cos θ + y sin θ, x sin θ − y cos θ) .
Thus T is the parallel translation of R2 that takes (x0, y0) to the origin, and R
is the reflection of R2 in the line through the origin that makes an angle of θ/2
radians with the positive x-axis. (You do not need to verify these facts.)
(a) Write down the rules for T−1 and R−1. You will get full credit if you do this
correctly, and you do not need to provide any justification.
Let C be some general curve in R2 defined by the equation
f(x, y) = 0 ,
where f(x, y) is some algebraic expression involving x and y, that is,
C = { (x, y) ∈ R2 | f(x, y) = 0 } .
It follows quickly that if B : R2 → R2 is any bijection then the image B(C) of
the curve C under B is defined by the equation
f(B−1(x, y)) = 0 .
(You do not need to verify this fact.)
(b) Deduce from part (a) that T (C), the image C under T , is the curve defined
by the equation
f(x+ x0, y + y0) = 0
and R(C), the image of C under R, by the equation
f(x cos θ + y sin θ, x sin θ − y cos θ) = 0 .
(c) Find an equation that defines the curve
T−1(R(T (C))) ,
that is, the curve that results as the image of C after first applying T , then
applying R and finally applying T−1.
(d) Let a, b, c ∈ R. Use your answer to part (c), or otherwise, to find an equation
for the line that results by reflecting the line with equation
ax+ by = c
in the line that passes through the point (x0, y0) making an angle of θ/2
should be in the form
a′x+ b′y = c′
for some real numbers a′, b′ and c′. 