Finite index and preservation of finiteness conditions for groups, semigroups and rings

Presenter: Nik Ruskuc (University of St Andrews)

Title: Finite index and preservation of finiteness conditions for groups, semigroups and rings

When/where: 15:00, Fri 3 Nov 2017, EN Fishbowl (EN.1.45, Parra campus), Western Sydney University

Abstract: It has become a folk-lore in group theory that subgroups of finite index share many basic properties with their parent group. For instance, Schreier's Theorem asserts that a subgroup H of finite index in a group G is finitely generated if and only if G is finitely generated. One can replace 'finitely generated' by 'finitely presented' (Reidemeister-Schreier Theorem), 'residually finite', etc. Motivated by the generality, ubiquity and importance of these types of results, one may be tempted to look for analogues in other algebraic systems. In this talk I will concentrate on such analogues for the Schreier and Reidemeister-Schreier theorems in rings and semigroups. It turns out that for rings both theorems carry over, and generalise to K-algebras over noetherian commutative rings. However, the proof exhibits some significant differences from the original group result. For semigroups, even the meaning of 'index' comes into question, but then it turns out that one can prove several different types of the above results, e.g. for subgroups of semigroups, or for Schutzenberger groups, or for subsemigroups of finite complement. I will conclude with some open problems arising from the comparison between rings and semigroups. The ring version of the Reidemeister-Schreier Theorem is recent joint work with Peter Mayr.