Pseudo Frobenius numbers

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 $a\ge 0$.

6 pages, expository, to appear in Expo. Math