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I'm preparing for the GRE Math Subject test, but I'm at a bit of a disadvantage, having never taken an algebra class or a probability class. Obviously I'm not going to learn a semester of abstract algebra in 3 weeks, but I'm hoping I can at least learn the basic definitions and memorize some of the most important basic results from these two fields so I'm not batting zero on these questions.

My question is: what results and definitions from a first semester of abstract algebra or probability theory would you say are indispensable?

I wouldn't object to a similar list for basic real analysis, but I actually have a background here; I just don't have a sufficiently strong background that I can necessarily pick out which parts were essential and which were just curiosities along the way.

Thanks!

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    $\begingroup$ In my experience on the algebra side of things the GRE likes to ask questions like "Which of these is/isn't a group/ring/field"? It would help if you were comfortable with these types of questions. Proficiency in this though (beyond just knowing the definitions), only comes from being familiar with all of these structures. $\endgroup$
    – wgrenard
    Aug 28, 2017 at 14:26

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For Abstract Algebra: Definitions of a group, Fermat's Little Theorem, knowing the order of groups and elements of groups, definitions of cyclic and abelian groups, and knowing what an ideal is and definitions of rings and fields.

For probability: Calculating Mean, standard deviation, and variance from probability density formula, as well as probability spaces and the probability of events occurring using Bernoulli trials.

For basic real analysis: Understanding limits, continuity, boundedness, compactness, and additional basic definitions of topological spaces, Lipschitz continuity, and sequences and series of functions.

It's really tough efficiently managing time on the calculus questions, I would focus on being able to do problems quickly so you have time for some of the more obscure questions that you inevitably encounter later in the exam, roughly past problem $44.$ I had a great deal of difficulty managing my time, so I suggest grilling properties of functions and calculus and not worrying as much on the abstract algebra and probability components, although you'll undoubtably have to study those rigorously to get a high score.

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    $\begingroup$ +1 for mentioning time management. I'd say this is the most difficult test I've ever taken simply because of the combination of time allotted and potential topics. To OP, I recommend thoroughly reading the prep book available on ETS's website, especially the part about the approximate breakdown of what's covered on the exam. Oh, and the practice test. Also there are a few other practice tests scattered around the net. $\endgroup$
    – user307169
    Aug 28, 2017 at 14:28
  • $\begingroup$ @tilper I'm currently looking for programs that may put less emphasis on the result of this exam, as my ability to do multiple problems quickly (roughly 2.5 minutes per question) is seriously limited. However, I will be taking this exam again, if only for a self-esteem boost from my first trial run, in which I took roughly half an hour for one improper integral-absolute value question and ran out of time later on. $\endgroup$ Aug 28, 2017 at 14:33
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    $\begingroup$ That's my fear as well. I'm definitely no speed demon. $\endgroup$
    – BenL
    Aug 28, 2017 at 15:11
  • $\begingroup$ @BenL It's a unique test in that there is a hodgepodge of information related to multiple disciplines...tests in college usually have one subject as you know, so its very hard to switch gears and work on different subjects $\endgroup$ Aug 28, 2017 at 17:08

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