Kourovka Notebook Problem 17.57

Definitions and mathematical statement

For each positive integer $m$ and integer $r$ with $0\le r<m$, define the residue class \[ r(m)=\{r+km:k\in{\mathbb{Z}}\}. \] For disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, define their class transposition $\tau_{r_1(m_1),r_2(m_2)}\in\operatorname{Sym}({\mathbb{Z}})$ by \[ \tau_{r_1(m_1),r_2(m_2)}(n)= \begin{cases} r_2+km_2,&n=r_1+km_1,\ r_1+km_1,&n=r_2+km_2,\ n,&n\notin r_1(m_1)\cup r_2(m_2), \end{cases} \qquad(k\in{\mathbb{Z}}). \] Let \[ G=\mathrm{CT}({\mathbb{Z}})=\left\langle \tau_{r_1(m_1),r_2(m_2)}: \begin{gathered} m_1,m_2\in{\mathbb{Z}}_{>0},\quad 0\le r_i<m_i\ (i=1,2),\\ r_1(m_1)\cap r_2(m_2)=\varnothing \end{gathered} \right\rangle \le\operatorname{Sym}({\mathbb{Z}}). \] The supplied source states that $G$ is simple and cites Math. Z. 264, no. 4 (2010), 927--938 for this background assertion.

Let $\rho\in\operatorname{Sym}({\mathbb{Z}})$ be the reflection $\rho(n)=-n-1$, and let $\alpha\in\operatorname{Aut}(G)$ be the automorphism induced by conjugation, \[ \alpha(\sigma)=\rho^{-1}\circ\sigma\circ\rho\qquad(\sigma\in G). \] Write $\operatorname{Inn}(G)$ for the subgroup of inner automorphisms, set $\operatorname{Out}(G)=\operatorname{Aut}(G)/\operatorname{Inn}(G)$, and denote the image of $\alpha$ in this quotient by $[\alpha]$.

Question (S. Kohl, 2010). Is \[ \operatorname{Out}(\mathrm{CT}({\mathbb{Z}}))=\langle[\alpha]\rangle\cong\mathrm C_2, \] where $\mathrm C_2$ is the cyclic group of order two? Equivalently, is $\alpha$ non-inner, and does every $\beta\in\operatorname{Aut}(G)$ admit a representation \[ \beta=\iota_g\circ\alpha^{\varepsilon} \quad\text{for some }g\in G\text{ and }\varepsilon\in\{0,1\}, \qquad \iota_g(x)=gxg^{-1}? \] The source's generator notation is interpreted as the outer class of the displayed conjugation automorphism. The supplied registry labels the problem open; this label is not an independently verified literature assessment.

Short English statement

S. Kohl asks whether every automorphism of the group generated by class transpositions of ${\mathbb{Z}}$ is an inner automorphism composed with either the identity or conjugation by $n\mapsto -n-1$, and whether this reflection induces a nontrivial outer automorphism. The supplied open status requires independent verification.

Sources

• Local UnsolvedMath registry: permanent ID $701359$, exact source code catalogue entry, title Kourovka Notebook Problem 17.57. The supplied record attributes the problem to S. Kohl (2010) and labels it open; its update date is 2026-10-16.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 17.57, 17th issue (2010); also identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• Background simplicity reference, as supplied: Math. Z. 264, no. 4 (2010), 927--938. The record does not supply the article title or authors.