论文标题
完整映射和正编形的构图和均等
Compositions and parities of complete mappings and of orthomorphisms
论文作者
论文摘要
我们确定置换组$ p _ {\ mathrm {comp}}（\ mathbb {f} _q），p _ {\ mathrm {orth}}（\ \ mathbb {f} _q _q）有限字段$ \ mathbb {f} _q $的正常形态等于$ \ peripatatorName {sym}（\ m athbb {f} _q）$，除非$ q \ in \ in \ in \ {2,3,4,5,5,8 \} $。更笼统地，用$ p _ {\ mathrm {comp}}（g）$表示，分别$ p _ {\ mathrm {orth}}}（g）$，$ \ operatatorname {sym}（s sym}（g）$的子组分别由完整的mappings产生，分别是由Orthomorphism $ g $ g $ g $ g $ g $。利用Eberhard-Manners-Mrazović和Müyesser-Pokrovskiy的最新结果，我们表明，对于每个具有完整映射的足够大的有限组$ G $（即，其Sylow $ 2 $ -SubGroups的sylow $ 2 $ -Subgroups是微不足道或noncyclic），是微不足道的或nontoncyclic），是$ p _ {\ mathrm {\ mathrm {comp} $ { $ p _ {\ mathrm {orth}}}（g）\ geq \ operatotorname {alt}（g）$。我们还证明$ p _ {\ mathrm {orth}}}}（g）= \ operatatorName {sym}（g）$，每一个足够大的有限解决的组$ g $具有完整的映射。证明这些结果需要我们研究完整映射和正编形的均等。还讨论了一些与密码学的已知结果以及与拉丁正方形的奇偶校验类型的连接。
We determine the permutation groups $P_{\mathrm{comp}}(\mathbb{F}_q),P_{\mathrm{orth}}(\mathbb{F}_q)\leq\operatorname{Sym}(\mathbb{F}_q)$ generated by the complete mappings, respectively the orthomorphisms, of the finite field $\mathbb{F}_q$ -- both are equal to $\operatorname{Sym}(\mathbb{F}_q)$ unless $q\in\{2,3,4,5,8\}$. More generally, denote by $P_{\mathrm{comp}}(G)$, respectively $P_{\mathrm{orth}}(G)$, the subgroup of $\operatorname{Sym}(G)$ generated by the complete mappings, respectively the orthomorphisms, of the group $G$. Using recent results of Eberhard-Manners-Mrazović and Müyesser-Pokrovskiy, we show that for each large enough finite group $G$ that has a complete mapping (i.e., whose Sylow $2$-subgroups are trivial or noncyclic), $P_{\mathrm{comp}}(G)=\operatorname{Sym}(G)$ and $P_{\mathrm{orth}}(G)\geq\operatorname{Alt}(G)$. We also prove that $P_{\mathrm{orth}}(G)=\operatorname{Sym}(G)$ for every large enough finite solvable group $G$ that has a complete mapping. Proving these results requires us to study the parities of complete mappings and of orthomorphisms. Some connections with known results in cryptography and with parity types of Latin squares are also discussed.
