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Finite Groups with Few Almost Maximal Subgroups
Abstract
Let G be a finite group. L. Morgana, C. E. Praeger, A.E. Zalesski \cite {MOR} introduced the notion of almost maximal subgroup of G. A subgroup H of G is said to be almost maximal in G if NG(H) is maximal in G. In this paper, let AM(G) denote the number of conjugacy classes of all almost maximal subgroups of G. First, we prove that G is Dedekindian if and only if AM(G)=0. Second, we show that the finite groups with AM(G)≤ 1 are supersolvable. Furthermore, we characterize the structure of finite groups with AM(G)=1. Final, we investigate a lower bound of AM(G) for certain non-solvable groups.
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BibTeX
@misc{meng2026finite,
title = {{Finite Groups with Few Almost Maximal Subgroups}},
author = {Wei Meng and Boru Zhang and Jiakuan Lu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02532},
eprint = {2611.02532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02532},
}
RIS
TY - UNPB
TI - Finite Groups with Few Almost Maximal Subgroups
AU - Wei Meng
AU - Boru Zhang
AU - Jiakuan Lu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02532
UR - https://arxiv.org/abs/2611.02532
AB - Let G be a finite group. L. Morgana, C. E. Praeger, A.E. Zalesski \cite {MOR} introduced the notion of almost maximal subgroup of G. A subgroup H of G is said to be almost maximal in G if N_G(H) is maximal in G. In this paper, let AM(G) denote the number of conjugacy classes of all almost maximal subgroups of G. First, we prove that G is Dedekindian if and only if AM(G)=0. Second, we show that the finite groups with AM(G)≤ 1 are supersolvable. Furthermore, we characterize the structure of finite groups with AM(G)=1. Final, we investigate a lower bound of AM(G) for certain non-solvable groups.
ER -