Kourovka Notebook Problem 21.19

Definitions and mathematical statement

Work with definable groups and maps in a common sufficiently saturated first-order structure; definability allows parameters. Morley rank is the rank with $\operatorname{RM}(X)\ge0$ for nonempty definable $X$, and $\operatorname{RM}(X)\ge k+1$ when $X$ contains infinitely many pairwise disjoint definable subsets of rank at least $k$. A group has finite Morley rank when the rank of its underlying definable set is finite. It is connected if it has no proper definable subgroup of finite index.

Let $S$ be an infinite simple group of finite Morley rank, and suppose every proper definable subgroup of $S$ is nilpotent. Simplicity means that the only normal subgroups are $1$ and $S$. Nilpotence means that the lower central series, obtained by repeatedly taking commutators with the group, reaches $1$ after finitely many steps. Let a nontrivial connected group $M$ of finite Morley rank act on $S$ by a definable map $M\times S\to S$, with each element inducing an automorphism. Assume the action is faithful, so only $1\in M$ fixes every element, and irreducible, so no definable subgroup $1<H<S$ is invariant under all of $M$.

Must this action be equivalent to the conjugation action of $S$ on itself? Explicitly, the required conclusion is that there are group isomorphisms $\theta:M\to S$ and $\eta:S\to S$ intertwining the actions: \[ \eta(m\cdot s)=\theta(m)\eta(s)\theta(m)^{-1} \quad(m\in M,\ s\in S). \] Thus identifying the acting group abstractly is insufficient without identifying its action. The irreducibility hypothesis refers to all definable subgroups, not only normal or connected ones.

Short English statement

Must a faithful irreducible definable action on an infinite simple finite-Morley-rank group with nilpotent proper definable subgroups be its conjugation action?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2528 (KOU-21.19), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.19. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.