Tools for understanding Abstract Mathematics

Question

I want to explore the concepts of Groups, Rings and Fields for practical application.

1. Are there beginner-friendly tools available to simplify, accelerate, verify the understanding of concepts in Abstract mathematics?

2. Are these tools publicly available? free of cost?

3. Do they provide some kind of Visualization?

My interests are in Abstract Algebra, but I also want to use a generalized tool that can handle other pure mathematics like Geometry, Number Theory and Mathematical Analysis.

_{Preferred Tools: Software that can run on Android (or) Ubuntu (or) Windows}

Answer 1

I highly doubt that it is possible the make a beginner-friendly, simplifying, accelerating, verifying, visualizating etc... tool for practical application. You still have some work to do in any way. And some things are not made to be visualized (well I do not stop you to try) but even after a while of studying it I do not have a clue how to visualize e.g. ideals of rings or normal subgroups etc. Of course you get a feeling for everything you work with - provided you do it for long and deep enough.

But the following programs are free and I think might help you.

• For group theory I recommend GAP
• As general CAS I can recommend WxMaxima
• For numerical purposes and matrix calculations there is Octave (Make sure to get GNUplot)
• A very limited software used for visualizations in algebraic geometry is SURFER
• For statistical purposes (even just fiddling around) R is the way to go.
• For 2D geometry and Plotting curves and playing with parameters I highly recommend Geogebra
• I did not have experience with it myself, but I heard a lot of good things about Sage

Answer 2

Limited to groups (and fairly basic at that), but pretty visual is Group Explorer.

I think it covers groups up to normal subgroups and homomorphisms, but it's been a while since I played with it. It runs on Windows (and Linux if you know how to compile stuff).