Weekly Essentials 8
MAT301
1 Academic Integrity
Please review the academic integrity policy and guidelines for the course. We are now more
than halfway through the semester and I will not accept any excuse for the use of AI to solve
problems in the course. If there are resources that you would like to use that are not on the
course list, please ask before you use them. I will no longer provide you any chances to put
away outside resources on quizzes.
Violations of academic integrity guidelines is not fair to your classmates, it is not fair to
your own learning, and it degrades the value of a UofT degree. It is my responsibility to
set guidelines that will ensure you learn, to monitor the following of those guidelines, and to
refer any suspected cases to the experts to investigate and to adjudicate.
2 How do you know what to assume?
This week you will be doing more and reading less.
One part of proof is knowing what you can assume; in fact that is always our starting point
as provers: clearly stating down what we are assuming or what we know is true, and what
we are being asked to find.
So, as we go back to be more rigorous about our approach to groups: how do we know what
we know?.
The guided notes from Professor Margaret L. Morrow, reviewed by other mathematicians
and published in the Journal of Inquiry-Based Learning give you a good framework for going
back to study.
I will use Prof. Morrow’s notes for this homework. When you come to prove a statement in
the notes, only use the theorems that have been stated before within the notes. While you
can use (rigorous arguments using visual tools, you should not assume all of the statements
that you have learned from Visual Group Theory. The entire point of this homework is
to help you re-build the theory. I’ve selected the exercises that you did not complete (or,
perhaps did not complete rigorously) below.
This week we will focus on the set of notes that comes before cosets (although the curriculum
is in a much different order than we saw it originally!).
1
MAT301 Winter 2024 Prof. Mayes-Tang
2.1 Making a set of notes
I suggest making yourself a set of notes where you compile the rigorous, formal definitions
and results.
3 Proving Basic Properties of Groups
Prove the following statements from igorously, using only the statements placed before.1
You know groups well now, so the justifications should not be too difficult. Ensuring they
are rigorous will be the difficult part.
Proofs: Basic Properties of Groups
12. If If a ∈ G and n ∈ N then both (an)−1 and (a−1)n are unambiguous interpretations
in terms of the definitions already given (what are they?). Prove that these two are
in fact equal. This element will be written as a−n.
13. Prove that if G is a group, witih a ∈ G, then (a−1)−1 = a.
17. Prove that if ab = e then ba = e. Use this to prove that if G is a group, with a and b
in G and ab = e, then a is the inverse of b.
19. Prove right cancellation property.
20. Suppose that G is a group with a, b ∈ G. Prove that x = a−1b is the unique solution
to this equation. What does this say about equations formed out of group elements?
Or about groups formed from “algebras?
Proofs: Properties of New Groups
27. Prove that G×H is a group when G and H are groups.
35. If S is a subgroup of G and T is a subgroup of H, prove or disprove that S × T is a
subgroup of G×H
Proofs: Cyclic Groups
60. Suppose H is a subgroup of a group G with a ∈ G. Suppose n,m, q, r are integers
with n = mq + r. Prove that if an and am are both in H, then so is ar.
61. Prove every subgroup of a cyclic group is cyclic.
64. Suppose that n and m are integers with d = gcd(m,n). Use group theory to show
that there are integers s and t with sm+ tn = d. Apply this to the case where m and
1Throughout, G is assumed to be a group, and all lowercase letters will denote elements of G. I have
attempted to retype the exercises that I’d like you to prove here for your convenience, but check the notes
if something doesn’t make sense as typos do creep in.
2 ©No copying, sharing, or posting without written permission
MAT301 Winter 2024 Prof. Mayes-Tang
n are relatively prime.
Proofs: Mappings Between Groups
77. Prove that identities go to identities and inverses map to inverses under homomor-
phisms.
78. Prove that if there is an isomorphism from G to H and G is abelian, then H is abelian.
79. Prove that if there is an isomorphism from G to H and G is has an element of order
n, then H has an element of order n.
82. Prove that the image of an abelian group under a homomorphism is abelian.
83. Show that you can have a homomorphism from an abelian group to a non-abelian
group. Can you explain why this is possible?
84. Prove that the kernel of a homomorphism from G is a subgroup of G.
85. Prove that the image of a subgroup under a homomorphism is a subgroup.
86. Make and prove a conjecture that relates the order of an element in G and the order
of the image of that element.
87. Let ϕ : G→ H be a homomorphism. Then either prove or disprove that if G is cyclic
then so is H.
3 ©No copying, sharing, or posting without written permission

学霸联盟