There is a general GRE. We will not discuss this. This file talks about the Subject Mathematics GRE (meant for students going to graduate school in mathematics).
50% of the exam is calculus and its applications. 25% consists of elementary algebra, linear algebra, abstract algebra and number theory. The remaining 25% consists of others areas commonly studied by undergraduates. Therefore, a thorough review of calculus is very important.
You can click on a link to the GRE Subject Mathematics site which includes a practice booklet. Looking at the practice booklet, it appears that the most important classes to take beyond Math 11 - 14, 22, 52 and 53 are (in descending order of importance) Math 153, 154, 111, 105, 103, 122. There were single questions from Math 113, 175, 177.
Princeton Review has a book called Cracking the GRE Mathematics Subject Test, 4th Edition . Prof Schaefer has a copy of this which you can check out for two weeks.
ETS has a recent recent sample test and an older, easier sample test . The blog GRE PRACTICE has worked out solutions for problems from the older, easier sample test. There is also a short sample test .
The calculus questions cover pretty much everything we teach in Math 11 - 14. Here is a very small sample of things that appear on the exam from these courses: i) interval of convergence of a power series, ii) Green's theorem, iii) a continuous function on a closed, bounded set takes on maximum and minimum values, and iv) if f is continuous on [b,c] then there's a d in [b,c] such that the integral from b to c of f equals f(d)(c-b).
Much, though not all, of the 25% algebra is covered in our Math 52 and 53. An important source of review would be the theorems in Math 52 about finite groups and their subgroups. From Math 53, know about matrices and operations on them, vector spaces and their subspaces, eigenvalues and characteristic polynomials.
The literatures says that the additional topics in the remaining 25% include elementary topology of R (the reals) and R^n, properties of continuous functions, differentiability and integrability, general topology, complex variables, probability and statistics, set theory, logic, combinatorics, discrete mathematics, algorithms and numerical analysis. There were also some problems where you follow through an algorithm and describe the output; a student who completed CSCI 10 should have no problem with these.
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Last update 9/27/11 by E. Schaefer.