Pseudo Sylow numbers
arXiv:1812.08988 · doi:10.1080/00029890.2019.1528825
Abstract
One part of Sylow's famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer $n\equiv 1\pmod{p}$ occurs as the number of Sylow p-subgroups of some finite group. While Hall's proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.
6 pages, expository, to appear in Amer. Math. Monthly