Details
| Original language | English |
|---|---|
| Pages (from-to) | 200-206 |
| Number of pages | 7 |
| Journal | Expositiones mathematicae |
| Volume | 37 |
| Issue number | 2 |
| Early online date | 25 Oct 2018 |
| Publication status | Published - Jun 2019 |
| Externally published | Yes |
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 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.
Keywords
- Frobenius’ theorem, Number of p-subgroups, Sylow's theorem
ASJC Scopus subject areas
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In: Expositiones mathematicae, Vol. 37, No. 2, 06.2019, p. 200-206.
Research output: Contribution to journal › Article › Research › peer review
}
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 -