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A classification of finite simply reducible groups of order at most 2000
Abstract
We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over 𝔽2, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.
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BibTeX
@misc{luan2026classification,
title = {{A classification of finite simply reducible groups of order at most 2000}},
author = {Yongzhi Luan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17215},
eprint = {2609.17215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17215},
}
RIS
TY - UNPB
TI - A classification of finite simply reducible groups of order at most 2000
AU - Yongzhi Luan
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17215
UR - https://arxiv.org/abs/2609.17215
AB - We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over 𝔽_2, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.
ER -