This definitely looks like the area of expertise for SAGEMATH folks.
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Valerilia,
It looks like you have the basics covered in your proposal, but it would be
good to find someone that is familiar with group theory to comment more.
Maybe you can see what features are available in group theory in other
mathematic software and then show how you can implement these here.
I've given it more thought and have drafted a proposal. Could someone have
a look?
https://github.com/sympy/sympy/wiki/GSoC-2017-Application:-Group-Theory.
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