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- W4320468838 abstract "We determine the permutation groups $P_{mathrm{comp}}(mathbb{F}_q),P_{mathrm{orth}}(mathbb{F}_q)leqoperatorname{Sym}(mathbb{F}_q)$ 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 $qin{2,3,4,5,8}$. 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-Mrazovi'c and Muyesser-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 $P_{mathrm{orth}}(G)geqoperatorname{Alt}(G)$. 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." @default.
- W4320468838 created "2023-02-14" @default.
- W4320468838 creator A5030863883 @default.
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- W4320468838 date "2022-07-20" @default.
- W4320468838 modified "2023-09-29" @default.
- W4320468838 title "Compositions and parities of complete mappings and of orthomorphisms" @default.
- W4320468838 doi "https://doi.org/10.48550/arxiv.2207.09642" @default.
- W4320468838 hasPublicationYear "2022" @default.
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