TY - JOUR

T1 - Pseudo Frobenius numbers

AU - Sambale, Benjamin

N1 - Funding information: The author thanks the anonymous reviewer for pointing out a missing argument in the proof of Theorem A . This work is supported by the German Research Foundation (projects SA 2864/1-1 and SA 2864/3-1 ).

PY - 2019/6

Y1 - 2019/6

N2 - 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 pa for some a≥0. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number [Formula presented] 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 3a for any a≥0.

AB - 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 pa for some a≥0. Extending previous results on Sylow's theorem, we prove in this paper that every Frobenius p-number [Formula presented] 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 3a for any a≥0.

KW - Frobenius’ theorem

KW - Number of p-subgroups

KW - Sylow's theorem

UR - http://www.scopus.com/inward/record.url?scp=85056670470&partnerID=8YFLogxK

U2 - 10.1016/j.exmath.2018.10.003

DO - 10.1016/j.exmath.2018.10.003

M3 - Article

AN - SCOPUS:85056670470

VL - 37

SP - 200

EP - 206

JO - Expositiones mathematicae

JF - Expositiones mathematicae

SN - 0723-0869

IS - 2

ER -