21.76 (2026)

Let $\sigma = (\sigma_{ij})$, $1 \leqslant i \neq j \leqslant n$, be an irreducible elementary net (carpet) of order $n \geqslant 3$ over a field $K$ (see 19.48). The net $\sigma$ is said to be closed if the elementary net subgroup $E(\sigma)$ does not contain new elementary transvections. The net $\sigma$ is said to be completable if its diagonal can be supplemented with subgroups to a complete net. Completable elementary nets are closed. It is known that over fields of characteristic 0 and 2 there exist irreducible closed elementary nets that are not completable (V. A. Koibaev, Trudy Inst. Mat. Mekh. Ural Div. Ross. Akad. Nauk, 17, no. 4 (2011), 134–141 (Russian); V. A. Koibaev, Siberian Math. J., 62, no. 2 (2021), 262–266).

Do there exist irreducible closed elementary nets of order $n \geqslant 3$ over a field of odd characteristic that are not completable?