MAT301 Assignment 1
Due: 23:59 EST, Monday, May 22, 2023
Justify all steps in your solutions with explicit references to the results that you use. You may only use
results from lectures, tutorials, Chapters 0–2 in your textbook, and prerequisite courses. Your solutions will
be graded based on their completeness, correctness, and clarity.
You can submit individually or in pairs. Submissions of 3 or more people will receive a grade of 0 (for
all members) with no appeal.
Exercise 1. Let G be a group. Suppose that there exists g ∈ G and a, b ∈ Z such that ga = gb = e. Prove
that ggcd(a,b) = e. (Recall that gcd(0, 0) = 0.)
Exercise 2. Let G be a group. Suppose that there exist g, h ∈ G such that g4 = e and g3h = hg3. Prove
that gh = hg.
Exercise 3. Let G be a group. Prove that G is abelian if and only if for all x, y ∈ G, we have (xy)2 = x2y2.
Exercise 4. Let G be a finite group. Prove that there are an even number of elements x ∈ G such that
x2 ̸= e and x3 ̸= e.
Exercise 5. Let n ∈ Z with n ≥ 3. Find all elements z ∈ Dn such that zx = xz for all x ∈ Dn.
Exercise 6. Let G be a set and let · : G × G → G be an associative binary operation that satisfies the
following properties.
1. There exists an element e ∈ G such that for each a ∈ G we have a · e = a. In other words, e is a right
identity
2. For each a ∈ G there exists an element b ∈ G such that a · b = e. In other words, every element of G
has a right inverse.
Prove that (G, ·) is a group with identity e. Note: This is not how we defined groups! (Hint: First prove
that for each a ∈ G, every right inverse of a is also a left inverse of a. Be careful not to use any results about
groups.)