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


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

*Are these tools publicly available? free of cost?

*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
 A: 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).
A: 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.


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*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
