# Pseudo Frobenius numbers

## TL;DR

This paper investigates the properties of Frobenius p-numbers, establishing conditions under which they correspond to Sylow p-subgroups in finite groups and identifying specific pseudo Frobenius numbers.

## Contribution

It extends Sylow's theorem to characterize Frobenius p-numbers and introduces the concept of pseudo Frobenius numbers, providing new insights into subgroup enumeration.

## Key findings

- Every Frobenius p-number congruent to 1 mod p^2 is a Sylow p-number.
- 46 is a pseudo Frobenius 3-number, with no corresponding finite group.
- The results generalize previous subgroup enumeration theorems.

## Abstract

For a prime p, we call a positive integer n a Frobenius p-number if there exists a finite group with exactly n subgroups of order p^a for some $a\ge 0$. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number $n\equiv 1\pmod{p^2}$ is a Sylow p-number, i.e., the number of Sylow p-subgroups of some finite group. As a consequence, we verify that 46 is a pseudo Frobenius 3-number, that is, no finite group has exactly 46 subgroups of order 3^a for any $a\ge 0$.