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I am currently busy with an introduction to Abstract Algebra (undergraduate module) through distance learning, but I am really struggling to answer the questions in my textbook. I know what all the concepts mean and can name all the properties, but when I get a question I usually have no idea where to start. I can usually only do the first quarter of the questions and then after that I am lost. Any hints or tips on how to approach Abstract Algebra. Please don't judge.Thanks, any tips would help.

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    $\begingroup$ You need a supplementary text with lots of worked problems. Try them, peeking a little if you get stuck. Once you understand it, review it to try to get an understanding of why the solution used a given line of attack. $\endgroup$ – quasi Apr 5 '17 at 6:24
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    $\begingroup$ Don't forget to test claims that you are asked to prove. In other words, play "skeptic". Once you see, by testing examples, that the given claim appears to be true, the examples might give a clue as to why the claim must be true. $\endgroup$ – quasi Apr 5 '17 at 6:28
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Here is some advice from Joe Gallian that I found nice when I learned algebra for the first time. I usually give this to my students at the beginning of the semester and many have said they found this to be helpful. Much of this is related to his book on Abstract Algebra (which I think is a great first one to learn from) but I think a lot of the advice is good for any book.

http://www.d.umn.edu/~jgallian/advice.html

Hang in there, you start to get the hang of it as you go! It is like solving a puzzle, the challenge is half the fun :) Best of luck studying a beautiful part of mathematics.

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A general tip for solving problems is to only consider the things you know about, and possible consequences. Once you start to freely express this creation then you will see that the theorems and questions that are proved are not random, rather are intimately related to the axioms of the abstract group. To be concrete about what I mean is look at the integer(whole) numbers with addition and multiplication, it is a non-trivial consequence that when you multiply any number with zero you get zero, and a non-trivial corollary to this is that you can not divid by zero which mystifies most elementary school students. But when you realize that for any $a\in \mathbb{Z}$ $0\times a= (0+0)\times a=0\times a+0\times a$ the first equality is by axiom of additive identity, the second is because of distribution axiom. If you can prove that $0$ is the unique number such that $0=0+0$ from group axioms then you will immediately see that $0\times a=0$.

In summary I sincerely was baffled by the theorems proved in abstract algebra when I was told them at first, however I learned to accept the nature of abstract algebra being that which writes down a finite set of axioms for the relationship operators and deduce the natural consequences.

P.S. you should be curious why you can't divide by zero, well use the definition of multiplicitive inverse and the fact zero times any number is zero not one.

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