Corrigendum and addendum to "Transitive permutation groups where nontrivial elements have at most two fixed points"
Paula Hähndel, Rebecca Waldecker
TL;DR
This corrigendum corrects a flaw in the 2-structure analysis of finite groups acting with low fixity, reinforces the main classification result for finite simple groups with fixity $2$, and presents a compact restatement of a key theorem under the exact fixity condition. It establishes corrected structural lemmas, including metacyclic behavior of point stabilizers and Frobenius-type decompositions, and proves the refined classification: finite simple groups with fixity $2$ are precisely $PSL_3(4)$, $PSL_2(q)$, or $Sz(q)$ under the revised hypotheses. The work is complemented by GAP-based verifications and an appendix that uses the table of marks to enumerate fixity-$k$ actions, enhancing reproducibility and computational checks. Overall, the corrections preserve the core results, clarify proofs, and extend practical computational tools for analyzing low-fixity permutation actions.
Abstract
This article revisits earlier work by the second author together with Kay Magaard. We correct several little results and we briefly discuss why, fortunately, the errors hardly affect our main theorems and in particular do not affect the classification of simple groups that act with fixity 2. As an addition to the submitted article, this version also contains GAP code in a little appendix at the end.