Getting Help in Abstract Math Classes

Question

This semester I am taking abstract algebra, and I am finding the homework to be much more difficult than calculus or linear algebra. The book my class uses is Fraleigh's Abstract Algebra. Although I know there are much more difficult algebra texts out there, I find proofs of certain theorems in the book are very difficult to follow. Also I get stuck on many homework problems that are assigned.

I know the questions I have listed below may seem very silly, but for an undergraduate with "average" abilities, any advice would be appreciated.

1. What should I do when I get stuck?
2. What should I do if I cannot understand a proof of a theorem after spending an hour or two on it? Is that amount of time even sufficient at my level?
3. How do I raise my mathematical abilities so that I am able to read more difficult texts in algebra and analysis?

Answer 1

I agree with Gerry Myerson up to a point: your teacher is certainly your primary resource. Anyone with even moderate experience teaching abstract algebra to a general audience of math majors should expect that a significant number of students will have trouble and be prepared to offer help.

The instructor isn’t the only possible source of help, however. Getting together on at least a semi-regular basis with other students can be quite useful, so long as you’re careful to avoid being ‘carried’ by one or two very good students. Even middling students generally find that they can fill in at least some of the gaps in one another’s understanding, simply because they don’t all have exactly the same gaps.

Another possibility is looking at discussions of the same material in other texts, or in notes that you may find on the web, though I think that you may have to reach a certain minimum level before that’s likely to be useful.

(And if you’re unlucky enough to have an unhelpful instructor $-$ sadly, they do exist $-$ such alternatives may be necessary as well as desirable.)

As for your third question, I think that for now I’d not worry too much about it. Just concentrate on the abstract algebra: if you start getting the hang of it, you’ll probably find that the techniques of learning and the general way of approaching problems that you pick up in this course will to a considerable extent carry over to other theory-oriented courses.

Answer 2

1. Go see your teacher.

2. Go see your teacher.

Answer 3

repeatedly review.. Sometimes we will have an idea from examples in the book, try to find the examples as much as possible, so that you can familiar with the prove of theorem.

Answer 4

The best advice is already given: see your teacher during office hours or set up an appointment.

Some questions you should ask yourself, in order, constantly:

Q1. Am I able to recite the formal definition of all of the words that show up in this theorem? If not, I should study the definitions.

Q2. Am I able to provide concrete examples that satisfy the properties of each word that shows up? If not, I should study examples.

Q3. Am I able to provide concrete examples that DO NOT satisfy the properties of each word that shows up? If not, I should study counterexamples.

Take this simple theorem from abstract algebra (group theory): "Let $H$ be a nonempty finite subset of a group $G$. If $H$ is closed under the operation of $G$, then $H$ is a subgroup of $G$."

My answer to Q1:

• "$H$ is nonempty <--> $H$ has at least one element",

• "$H$ is finite <--> the number of elements it has is some natural number",

• "$G$ is a group <--> $G$ is a nonempty set that has an operation, the operation is associative, there is an identity element, and all elements have inverses.

• "subset of $G$ <--> every element in $H$ is also in $G$",

• "operation <--> a function that takes two inputs from a set to some unique output element",

• "closed under operation of $G$ <--> $G$'s operation always outputs an element of $H$ no matter what two elements you operate on from $H$",

• "$H$ is a subgroup of $G$ <--> $H$ is a subset of $G$ and is also itself a a group under the operation of $G$".

My answer to Q2:

• $\{0,1\}$ is nonempty

• $\{0,1\}$ is finite

• $(\mathbb{Z}_4,+)$, the integers modulo $4$ is a group

• $\{0,1\}$ is a subset of $\{0,1,2\}$

• The function $\xi: \{0,1\} \times \{0,1\} \rightarrow \{0,1\}$ where $\xi(0,0)=0$, $\xi(0,1)=1$, $\xi(1,0)=1$, and $\xi(1,1)=0$ defines an operation on $\{0,1\}$ (this is "addition modulo $2$").

• Interpreting $\{0,1,2,4\}$ as the integers modulo ($(\mathbb{Z}_4,+)$), the operation $+$ acting on the set $\{0,2\}$ outputs only elements of $\{0,2\}$.

• $(\{0,2\},+)$ is a subgroup of $(\mathbb{Z}_4,+)$ as above.

It would be good practice to answer Q3. Making this loop of checking yourself forces you to absorb the definitions and see why they exist in the first place.

If those questions were easy to answer for you, then I suggest identifying the hypothesis and conclusion of your theorem. Dig up counterexamples to the hypothesis of the theorem and figure out why they don't necessarily work (sometimes they may! Be careful here, you may need to tinker). This type of study will help you put the theorem in context and make proving it easier.