On Some Problems from the Kourovka Notebook

Abstract

The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1\le k\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic ${\sum }_g\ddot{I}†(|g|)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.

21 pages. Lean 4 formalisation: https://github.com/pitmonticone/Kourovka. v2: Expands the appendix with an account on problem selection and adds the MathOverflow provenance of Problem 18.50