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Solvable Supplements to Normalizers of Cyclic 2-Subgroups
Abstract
Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by SSN2, sharply restricts the nonabelian composition factors of the group to the family PSL2(q), where q≥7 is a prime power satisfying q≡3pmod4. Conversely, this family is precisely the nonabelian finite simple groups that satisfy SSN2. Consequently, a finite group satisfying SSN2 is solvable if and only if it has no section isomorphic to one of these groups.
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BibTeX
@misc{qiao2026solvable,
title = {{Solvable Supplements to Normalizers of Cyclic 2-Subgroups}},
author = {Shou Hong Qiao and Binzhou Xia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14929},
eprint = {2609.14929},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14929},
}
RIS
TY - UNPB
TI - Solvable Supplements to Normalizers of Cyclic 2-Subgroups
AU - Shou Hong Qiao
AU - Binzhou Xia
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14929
UR - https://arxiv.org/abs/2609.14929
AB - Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by SSN_2, sharply restricts the nonabelian composition factors of the group to the family PSL_2(q), where q≥7 is a prime power satisfying q≡3pmod4. Conversely, this family is precisely the nonabelian finite simple groups that satisfy SSN_2. Consequently, a finite group satisfying SSN_2 is solvable if and only if it has no section isomorphic to one of these groups.
ER -