## Mathematical Ideas

Working on triangles in groups (different from triangle group) makes me explore in fact some kind of symmetries in non abelian groups. If you google this subject (symmetries in non abelian groups), you will find that not a lot of references exist. I found a book “Inverse semigroups: the theory of partial symmetries” which may be interesting. I actually don’t know if this book is interesting or not, but I ordered it on amazon, and I will see. It will make grow from one unit my personal library which counts more than 100 books on groups, semigroups, fields and other fundamental algebraic structures in the middle of 300 other books on other mathematical subjects.

In fact, after having found something possibly interesting (object, property or conjectures), I immediately search for articles or books on the same subject or on subjects which are linked one way or another with the idea I have. This is the most fundamental lesson you learn when you do a PHD: “study what has been done before you”. Know who has written what, and read it.

Sometimes, the idea I have, has been explored nicely and completely by others. Sometimes I do not find any article, so I explore the concept the more I can, before discovering that the subject has been long studied and that I didn’t know the terms (the vocabulary) to use to find the articles on this subject. Sometimes, I’m not able to explore the idea I have. It’s too complicated (at least for me). And sometimes it’s an original idea. Original does not mean great. Most of the time the idea I have, is not a great one. However when I understand it, I still enjoy having learned something, even it is unsignificant, even if it’s a tiny step, because it always indicates me where could be some other steps.

For example, triangles in groups are not a great idea, however I know when they occur (I demonstrated it) and most importantly when they do not occur. In this cases, can I find some other special symmetries ?

Symmetries are fascinating, they give a sort of order in disordered world. The best book on this subject for everyone and not for a specialist of this subject is “Symmetry: A Journey into the Patterns of Nature” from Marcus Du Sautoy. You don’t need to be a mathematician to understand this book. It is well written and can be really read like a novel.

Groups (as a mathematical object) are built upon symmetry. That’s why they are not so numerous. In fact they are very scarce. There are far more magmas than groups at a given order.