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!
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.