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Finite groups with exactly 2p + 1 subgroups of order p
Abstract
Let m_p(G) be the number of elements of order p in a finite group G, p prime. We prove that there is no finite simple group G with m_p(G) = 2p^2-p-1, and that a finite group with this property exists if and only if 2p+1 is prime or 3^f with f an odd prime. This concerns an open problem and extends the classification of finite groups with few elements of prime order.
Related
- Finite groups whose proper subgroups of order divisible by p are supersolvable
- Finite abelian subgroups of algebraic groups
- On second maximal subgroups of finite groups
- Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups
- Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions
BibTeX
@misc{chinyere2026finite,
title = {{Finite groups with exactly 2p + 1 subgroups of order p}},
author = {Ihechukwu Chinyere and Bernard O. Bainson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35643},
eprint = {2610.35643},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35643},
}
RIS
TY - UNPB
TI - Finite groups with exactly 2p + 1 subgroups of order p
AU - Ihechukwu Chinyere
AU - Bernard O. Bainson
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35643
UR - https://arxiv.org/abs/2610.35643
AB - Let m_p(G) be the number of elements of order p in a finite group G, p prime. We prove that there is no finite simple group G with m_p(G) = 2p^2-p-1, and that a finite group with this property exists if and only if 2p+1 is prime or 3^f with f an odd prime. This concerns an open problem and extends the classification of finite groups with few elements of prime order.
ER -