On Sufficient Conditions for the Closure of an Elementary Net

摘要

In the paper, the elementary net closure problem is considered. An elementary net (net without a diagonal) sigma = (sigma(ij))(i) (not equal) (j) of additive subgroups sigma(ij) of field k is called "closed" if elementary net group E(sigma) does not contain new elementary transvections. Elementary net sigma = (sigma(ij)) is called "supplemented" if table (with a diagonal) sigma = (sigma(ij)), 1 <= i, j <= n, is a (full) net for some additive subgroups sigma(ii) of field k. The supplemented elementary nets are closed. The necessary and sufficient condition for the supplementarity of elementary net sigma = (sigma(ij)) is the implementation of inclusions sigma(ij)sigma(ji)sigma(ij) subset of sigma(ij) (for any i not equal j). The question (Kourovka Notebook, Problem 19.63) is investigated of whether it true that, for closure of elementary net sigma = (sigma(ij)) it suffices to implement inclusions sigma(2)(ij)sigma(ji) subset of sigma(ij) for any i not equal j (here, (sigma(2)(ij) denotes the additive subgroup of field k generated by the squares from sigma(ij)). The elementary nets for which the latter inclusions are satisfied are called "weakly supplemented elementary nets." The concepts of supplemented and weakly supplemented elementary nets coincide for fields of odd characteristic. Thus, the aforementioned question of the sufficiency of weak supplementarity for the closure of an elementary net is relevant for the fields of characteristics 0 and 2. In this paper, examples of weakly supplemented but not supplemented elementary nets are constructed for the fields of characteristics 0 and 2. An example of a closed elementary net that is not weakly supplemented is constructed.