I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics). 

I would likely use it for group theory, number theory, algebraic geometry and probably polytopes. 

The kind of program I have in mind is Mathematica or Matlab. Although probably those are not designed for abstract mathematics.

Any suggestions?

Here are some links.


Axiom and Maxima are good general purpose computer algebra systems.
DataMelt is a free Java-based math software with a lot of examples
GAP is a system for computational discrete algebra (with particular emphasis on computational group theory).
PARI/GP is a CAS for fast computations in number theory.
SAGE is a kind of unified framework for several systems, including GAP, PARI, and Maxima.
Octave is a system for numerical computations (it is close to Matlab).
Cadabra is a computer algebra system designed for the solution of the field theory problems.
CoCoA stands for "Computations in commutative algebra".
KANT / KASH stands for "Computational Algebraic Number Theory".
Macaulay 2 is a system for research in commutative algebra and algebraic geometry.
Snap is a computer program for studying arithmetic invariants of hyperbolic 3-manifolds.
Symmetrica is an object oriented computer algebra system for representations, combinatorics and applications of symmetric groups.

