Monday 19 January 2026 at 11:30 am, as part of the Al@Bicocca seminar cycle, Martino Garonzi (University of Ferrara) will give the following talk
Title: The Herzog-Schönheim Conjecture for finite simple groups
Abstract: In the 1950’s Davenport, Mirsky, Newman and Rado proved that if the integers are partitioned by a finite set of arithmetic progressions, then the largest difference must appear more than once. In other words, if g1,..., gn and a1 ≤ a2 ≤ ... ≤ an are integers such that {gi + aiZ}ni=1 is a partition of Z then an−1 = an. This confirmed a conjecture of Erdos and opened a broad area of research (see Covering systems of Paul Erdös. Past, present and future, Paul Erdös and his mathematics, I (Budapest, 1999), Bolyai Soc. Math. Stud., vol. 11, pp. 581- 627. Janos Bolyai Math. Soc., Budapest (2002) for a detailed bibliography). The Herzog–Schönheim Conjecture (1974) states that, if a group G is partitioned into cosets H1x1,..., Hnxn, then the indices |G : Hi |, i = 1,..., n, cannot be pairwise distinct. It is known that, in order to prove this conjecture in general, it is enough to prove it for finite groups. The conjecture holds for finite groups having a Sylow tower (Berger et al. 1987), so in particular for supersolvable groups. In this talk, I will present a proof of this conjecture for all finite simple groups and symmetric groups. This is a joint work with Leo Margolis (Universidad Autónoma de Madrid). A preprint of the paper is available at the following ArXiv link: https://arxiv.org/abs/2509.25118
Information to attend
The seminar will be held in Room U9-11 (U9 Building | Viale dell'Innovazione 10, Milano) and will also be available online by this link.
For more information, visit the website