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The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces
Abstract
Let X be a finite skew brace, and let L∞(X) be the final term of its left series. We introduce the left-nilpotent residual RL(X) and prove that $ \RL(X)=\Id_X\bigl(L_\infty(X)\bigr). $ Thus X/RL(X) is the largest left-nilpotent quotient of X. For skew braces of nilpotent type, we relate RL(X) to γ∞(X,·) and obtain conditions ensuring that L∞(X) is an ideal, including the case where |X| is cube-free. Our main results concern supplements to RL(X). We prove existence results using coprime action and Sylow and Hall theory, and give an example of order $18$ whose residual has no proper supplement. Finally, iterating RL yields the largest perfect subskew brace of X, and X is KSV-solvable if and only if this subskew brace is zero.
Related
- Inverse Baer deformations and finite simple skew braces of nilpotent type
- Classification of finite left-simple skew braces
- Classification of skew left braces with additive group isomorphic to the infinite dihedral group
- Skew braces whose left series is not generated by right-normed star products
- Left braces of size p²q² , for p and q odd primes
BibTeX
@misc{ercan2026leftnilpotent,
title = {{The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces}},
author = {Gülin Ercan and İsmail Ş. Güloğlu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06994},
eprint = {2609.06994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06994},
}
RIS
TY - UNPB
TI - The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces
AU - Gülin Ercan
AU - İsmail Ş. Güloğlu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06994
UR - https://arxiv.org/abs/2609.06994
AB - Let X be a finite skew brace, and let L_∞(X) be the final term of its left series. We introduce the left-nilpotent residual RL(X) and prove that $ \RL(X)=\Id_X\bigl(L_\infty(X)\bigr). $ Thus X/RL(X) is the largest left-nilpotent quotient of X. For skew braces of nilpotent type, we relate RL(X) to γ_∞(X,·) and obtain conditions ensuring that L_∞(X) is an ideal, including the case where |X| is cube-free. Our main results concern supplements to RL(X). We prove existence results using coprime action and Sylow and Hall theory, and give an example of order $18$ whose residual has no proper supplement. Finally, iterating RL yields the largest perfect subskew brace of X, and X is KSV-solvable if and only if this subskew brace is zero.
ER -