Compositions and parities of complete mappings and of orthomorphisms

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-Mrazovi\'c and M\"uyesser-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.