# Pseudo Sylow numbers

## TL;DR

This paper investigates the possible numbers of Sylow p-subgroups in finite groups, proving through elementary methods that certain numbers, like 35 for Sylow 17-subgroups, cannot occur.

## Contribution

It provides an elementary proof that specific integers, such as 35, cannot be the number of Sylow p-subgroups in any finite group, extending understanding beyond Hall's original results.

## Key findings

- No finite group has exactly 35 Sylow 17-subgroups.
- Certain integers congruent to 1 mod p are impossible as Sylow p-subgroup counts.
- Elementary methods suffice to prove these non-existence results.

## 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.