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I'm doing self study on a couple of topics in mathematics, such as real analysis, abstract algebra, and linear algebra. From time to time, there are always a couple of exercises which I found too difficult to solve. I spend quite some time to think about them. When I fail, I google to find the solutions. Most of the time, I get the solutions.

However, there are some downsides to this attitude. First, I would spend too much time on a single problem. So I feel that my progress is a little bit too slow. Other than that, when I read a solution, I don't get the real understanding of the problem. When I read a proof, my brain is working mechanically to check every statement, so I don't know what exactly is going on.

I start to wonder whether I'm doing this correctly. What I want to ask here is, what should I do when I encounter difficult exercises? Should I think about them myself until I get the answers? Or should I skip the difficult ones and move on to the next chapters, then go back after I gain more understanding?

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    Depends on the exercises. Can you give examples?2011-11-03
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    I'm working on Herstein's _Topics in Algebra_2011-11-03
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    I mean can you give examples of exercises you get stuck on?2011-11-03
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    Here are some of them: -If $G$ is a group and $H,K$ are two subgroups of finite index in $G$, prove that $H\cap K$ is of finite index in $G$. -If an abelian group has subgroups of orders $m$ and $n$, respectively, then show it has a subgroup whose order is the least common multiple of $m$ and $n$. -If $N$ is a normal subgroup in the finite group such that $i_G(N)$ and $o(N)$ are relatively prime, show that any element of $x\in G$ satisfying $x^{o(N)}=e$ must be in $N$. I've found the solutions now, though, except for the second one which seems to require something beyond what I've learned.2011-11-03
  • 3
    what a great question !2011-11-04
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    Thanks for all the helpful responses guys!2011-11-04

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