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Is it normal to be so discouraged by abstract algebra?

  1. Jan 25, 2013 #1
    I'm currently in my first abstract algebra course, focused on sets, groups, arithmetic modulo, rings, fields etc. Ive never taken an abstract course before. Ive taken:

    Calc 1-2
    Linear Algebra
    Advanced Applied Linear Algebra

    so the concept of abstraction is very new to me; I find that I don't understand much in class when I leave and at times I don't even want to study it frustrates me so much. The book is Keith Nicholson's Introduction to Abstract Algebra. The teacher is known to be VERY good, which makes me feel worse because it's not getting through to me very easily... and in class he'll say things like "I'm sorry we have to go through this I know most of you know this part or that part etc..."

    We just began groups after doing injection/surjection/bijection and equivalence relations, which I could explain to you, but never solve a question in practice about the equivalence relations. Anyone else feel like this? Is it just like other forms of math where I need to grind through problem after problem? Ive always been self conscious about my mathematical abilities so maybe i'm blowing it out of proportion...

    This is an example of the things im still struggling with, 3 weeks into algebra structures...
  2. jcsd
  3. Jan 25, 2013 #2


    Staff: Mentor

    Your difficulty may stem from not doing math proofs until now. In the old days, HS geometry would have a lot of proofs to do and you got used to determining what is given, the best style of proof to use for the problem at hand...

    i'd talk with your prof about it especially since you said he's an excellent teacher. He didn't get that honor by not helping students.
  4. Jan 25, 2013 #3
    I agree. As we speak I am going through a boatload of proof problems on set theory... and honestly I get it; at this level its just a manipulation of the set properties he gave us at the beginning of the year... I just don't have that feeling that I could do this on a test by myself.. I'm hoping I can get in with him next week; our schedules have diabolically clashed; but he does know I am struggling. It's difficult when the guys around me all took honours abstract linear/elementary analysis/foundations of mathematics/discrete.

    Let A, B, C and D four nonempty sets. If f : A → B and g : C → D are two func-
    tions, we define a new function h : A×C → B×D as follows: ∀(a, c) ∈ A×C, h(a, c) = (f(a), g(c)).
    Show that h is bijective if and only if f and g are bijective.

    For example this question... this is a question on an assignment I have due next week and I honestly don't have a clue how to go about it; I try to reference the text... definitely a no go. Normally I have an idea at least!
  5. Jan 25, 2013 #4


    Staff: Mentor

    Yes, you reminded me that at our school Set theory and Proof came before Abstract Algebra. I remember now that as a Physics student I was too ambitious and took Algebraic Topology passing barely (the kindness of the prof) then taking Abstract Algebra with a plan to read the Set Theory book while I was taking it. After that I had enough credits to graduate early which I did because of burnout (was working 30 hrs per week too). In hind sight I should've quit my job and focused on school but I just wanted out and wanted it paid off too.
  6. Jan 25, 2013 #5
    Does it move much slower/cleaner when one gets right into the abstract algebra? I feel like he kind of rushed through set theory and functions/equivalence relations assuming we knew a lot of it and today we just got into the axioms of group theory. Here is my course calendar:

    Contents: Arithmetic modulo n, permutations, groups, cyclic groups, homomorphisms, quotient groups, isomorphism theorems, rings, fields.

    I did a lot of modular arithmetic with things like the Euclidean algorithm and arithmetic on fields in my applied class, but I feel like it probably won't be much like that, haha.
  7. Jan 25, 2013 #6
    This seems like a rather "light" schedule. So they'll probably won't go very fast. That said, things will really go very abstract. Quotient groups and isomorphism theorem are not very easy concepts if you encounter them for the very first time and can seem really weird and unmotivated.

    I fear that you are going to go through some rough times in this class. Be prepared for that. Try to use all the resources you have: office hours, study groups with fellow students, a tutor, this forum.

    I can assure you that the work pays off. Many people struggle very much when they encounter proofs and abstract algebra. But eventually, it will click and things will be easy. Just put in the effort.
  8. Jan 25, 2013 #7
    I appreciate the input. I will definitely be using anything and everything; this class is very important to me... like I said, I have always lacked math confidence; i'm not a particularly great student, I struggled through most maths.. I am hoping that passing this will solidify in my mind that I CAN do it. Ive fulfilled the entirety of my BSc economics part (im a double major) so it's all math from here on out.

    I am lucky he does post a very detailed list of problems with solutions for each chapter... but the assignment im currently working on I don't really know how to approach and that is the most distressing thing for me. I did all of his suggested exercises on set theory and I still have trouble with whether or not im correct on the the 2 questions on it for the assignment which tells me something may not be clicking.
  9. Jan 25, 2013 #8


    Staff: Mentor

    You may be succumbing the fear of failure and as such aren't allowing yourself to relax and look at the problem objectively. Whats important in proofs is to see the way thru the proof and then to do the steps. To learn that you need to look at known proofs and ask yourself why did they set it up that way and proceed from there.

    There's a good book on proofs of all types called: Proofs from The Book which was inspired by Mathematician Paul Erdos who would always say my brain is open and often said this elegant proof must be in The Book.

    Last edited by a moderator: May 6, 2017
  10. Jan 25, 2013 #9
    Proofs from the Book is not a good introduction (it's not a textbook at all, actually; it's just a collection of some particularly elegant proofs). Unless the OP has experience with analysis, algebra, and combinatorics, he won't get very far with it.
    Last edited by a moderator: May 6, 2017
  11. Jan 25, 2013 #10
    I believe you're right; I build it up very much in my head because I think if I struggled through calculus how would I ever do higher math. Definitely needs addressing. My father actually has that book btw haha.
    Last edited by a moderator: May 6, 2017
  12. Jan 25, 2013 #11


    Staff: Mentor

    There you go, you were born to do math.
  13. Jan 25, 2013 #12
    Unfortunately, getting used to abstract algebra proofs just takes time and practice. Trying to cram all the background in too quickly is always a source of frustration. If you want an alternate reference, for a little extra insight I often recommend Pinter's "A Book of Abstract Algebra," which is now an inexpensive Dover paperback.

    Keep at it. Algebra is a beautiful subject once you get past the first hurdles.
  14. Jan 26, 2013 #13
    Everyone's first proof-based class is a struggle (unless you took a "transition to proofs" class or whatever). I took a proof-based linear algebra class as my first which I thought was a better choice since I had used and seen some of the concepts before, but this time we were doing them more rigorously. So I could see the point. This is in contrast with abstract algebra where things can seem very unmotivated. I took it the semester after, and it still took some time to click, but once it did, I actually found the class fairly easy. So my advice would be to just stick with it, try to understand the theorems and proofs and then do lots of exercises.

    I second this recommendation. It's a good book that doesn't assume much mathematical maturity so it should suit you pretty well.
  15. Jan 26, 2013 #14
    I appreciate the insight. I have talked to a couple classmates who took a course "Foundations of Mathematics" which was all proofs and they say they find the proof-based courses a little easier because it all has a familiar feel; i'm hoping for a similar situation as I enter the spring semester when I'll be taking analysis. Thanks again
  16. Jan 26, 2013 #15
    Well if your class will be using Rudin good luck :D. It will definitely help and yes, you'll be familiar with how proofs work and coming up with them in basic situations, but Analysis proofs are very different. I suspect you'll have an easier time in a proof based linear algebra class though, since algebra and LA proofs have somewhat of a similar feel. But single-variable analysis sucks. Multivariable is a lot cooler.
  17. Jan 26, 2013 #16
    This is certainly a very good recommendation.

    In science and mathematics, you will hit a wall sooner or later. Something that looks so difficult that you have no idea how you're going to understand it. I first hit a wall when I read about Cantor and his set theory, it was very confusing to me. But if you put in the effort, then eventually things will click and things will become obvious.

    Don't feel bad, everybody hits a wall. But it's only the real scientists who put in the effort and who eventually break the wall.
  18. Jan 26, 2013 #17
    What you have to notice about proofs in abstract algebra is they're VERY systematic!

    For example, if you need to prove something is a group what do you do? You need to prove that it is closed, has an inverse, has an identity, and is associative.

    For closure, the proofs will always be in a format similar to:

    Let x,y be in G. Hence, <insert what it means to be an element in G>. <Algebra to show xy is in G> Therefore, xy is in G and thus, G is closed.

    The hard part is the middle, but that's just manipulating algebra. Just as for closure, identity, inverse and associativity all are systematic.

    Good luck.

    EDIT: I don't how far you're in abstract, but xy is WRT to the operator of the group.
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