For a finite group group of prime power order which can be defined minimally by d generators and r relations, the Golod-Shafarevich theorem asserts that $r> d ^ 2/4$. This has important consequences in number theory. After recalling the whole context, I will present a generalization of the GS theorem, and some new consequences of this generalized version. We will take the time to present the arithmetic objects, so that the lecture is intended for a general audience.

4T relations of chord diagrams are very useful in computing Vassiliev invariant of a knot. These relations can be purely put in the language of graph theory. Stanley’s chromatic symmetric function of the graph satisfies a variant of these relations. We will describe these ideas and see the relevant notions in the matroid theoretic setting.