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Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups
Abstract
A complete mapping of a finite group G is a permutation θ of G such that x↦ xθ(x) is also a permutation of G. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every n≥2, every prime power q, and every Z≤ Z(SLn(q)), the group SLn(q)/Z admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if (n,q)≠(2,2).
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BibTeX
@misc{wang2026complete,
title = {{Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups}},
author = {Shixin Wang and Shikang Yu and Tao Feng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17631},
eprint = {2610.17631},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17631},
}
RIS
TY - UNPB
TI - Complete mappings stabilizing Sylow $2$-subgroups in central quotients of special linear groups
AU - Shixin Wang
AU - Shikang Yu
AU - Tao Feng
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17631
UR - https://arxiv.org/abs/2610.17631
AB - A complete mapping of a finite group G is a permutation θ of G such that x↦ xθ(x) is also a permutation of G. Complete mappings have applications in cryptography, particularly in the design of round functions for cryptographic systems. The study of their parity has led to the question of which finite groups admit complete mappings of both parities. A useful approach to this question is to construct complete mappings stabilizing suitable subgroups, in particular Sylow $2$-subgroups. We show that, for every n≥2, every prime power q, and every Z≤ Z(SL_n(q)), the group SL_n(q)/Z admits a complete mapping stabilizing a Sylow $2$-subgroup if and only if (n,q)≠(2,2).
ER -