Kourovka Notebook Problem 21.52

Definitions and mathematical statement

Let $L$ be a finite nonabelian simple group and let $D$ be one conjugacy class of involutions, meaning elements of order two. Form the complete simple graph on vertex set $D$, and label its unordered edge $\{a,b\}$ by the integer $o(ab)$, the order of the product. This is well defined on unordered pairs, since $ba=(ab)^{-1}$ for involutions and inverse elements have the same order.

Its colour-preserving automorphism group is \[ A=\{\tau\in\operatorname{Sym}(D): o(\tau(a)\tau(b))=o(ab)\text{ for all distinct }a,b\in D\}. \] The colours are the actual integer orders; permuting colour labels is not allowed. Let $\operatorname{Aut}(L)_D$ be the subgroup of group automorphisms mapping $D$ to itself. Restriction to $D$ gives a homomorphism $\operatorname{Aut}(L)_D\to A$.

Does every $\tau\in A$ extend to an automorphism of $L$? This is the precise interpretation of the proposed containment of graph automorphisms in $\operatorname{Aut}(L)$. The restriction homomorphism is injective: $\langle D\rangle$ is a nontrivial normal subgroup of the simple group $L$, and consequently is all of $L$.

Thus an extension, if it exists, is unique, and the target can be written $A=\operatorname{Aut}(L)_D|_D$. The task must hold for every choice of involution conjugacy class, not only for its union with other classes. Preservation of pair-product orders is the only graph data supplied.

Short English statement

Does every permutation of an involution conjugacy class preserving all pair-product orders extend to an automorphism of the finite simple group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2561 (KOU-21.52), 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 21.52. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.