Linear subspaces of finite fields with large inverse-closed subsets
Let \(F\) be a finite field (or an infinite field with characteristic distinct from 2) and let \(A\) be a nontrivial subgroup of \((F,+)\) such that \(\{ a^{-1} \mid 0 \neq a \in F \} \cup \{0\}\) is also an additive subgroup. The author proves that \(A\) is then a one-dimensional subspace over a subfield of \(F\). This is a corollary of more general results which say that an additive subgroup of \(F\) with a large inverse-closed subset is necessarily inverse-closed. Examples from finite geometry show that some bounds in these results are sharp. Related results are contained \textit{S. Mattarei} [Isr. J. Math. 159, 343--347 (2007; Zbl 1211.12007)], \textit{D. Goldstein} et al. [Pac. J. Math. 227, No. 2, 287--294 (2006; Zbl 1122.17024)] and \textit{H.-J. Kroll} [Geom. Dedicata 43, No. 1, 59--64 (1992; Zbl 0758.51004)].
scientific article; zbMATH DE number 6125338