Two colours do not determine the rest

Let $D$ be a conjugacy class of involutions in a finite nonabelian simple group $G$. Colour the edge joining distinct $a,b\in D$ by the order $|ab|$. I. B. Gorshkov asks whether preserving colours $2$ and $p$, where $2<p$ are the two smallest prime divisors of $|G|$, forces a permutation of $D$ to preserve every colour.

Theorem. The answer is negative. There is a finite nonabelian simple matrix group and a transposition of a whole involution class that preserves colours $2$ and $3$, but changes an edge of colour $5$ into one of colour $7$.

The construction uses Wilson's Ree matrix model over the field of eight elements. Its essential feature is a pair of involutions which are indistinguishable by commutation, yet distinguishable by other product orders.

Exchanging two indistinguishable vertices

Suppose that $x,y,a\in D$ satisfy

and that no product of two elements of $D$ has order three. Let $h$ exchange $x$ and $y$ and fix every other element of $D$.

For distinct involutions, product order two is equivalent to commutation. The centralizer equality therefore says that every vertex outside $\{x,y\}$ has a colour-two edge to $x$ precisely when it has one to $y$. The pair $\{x,y\}$ is itself preserved, and all other pairs are fixed. Thus $h$ preserves colour two. It also preserves colour three, since that colour does not occur. But $a$ is distinct from $x,y$, so

This proves the theorem once the three involutions and the two global properties have been established. In particular, the permutation acts on the entire conjugacy class, not merely on a selected subgraph.

The matrix construction

Work over $k=\mathbf F_2[\omega]/(\omega^3+\omega+1)$. Wilson's construction supplies explicit matrices in dimension $26$, a unipotent subgroup $U$, a diagonal torus $H$, and a group $W=\langle r,s\rangle$ of sixteen Weyl representatives. Their generated group $G$ has a Bruhat decomposition

The last positive root subgroup $Z=\{z(t):t\in k\}$ is elementary abelian and is centralized by $U$. Set $x=z(1)$ and $y=z(\omega)$. These are distinct involutions conjugate under $H$. A suitable Weyl conjugate $a$ of $x$ satisfies

These are finite-field matrix identities; since $5$ and $7$ are prime, they give the required exact orders.

Two further matrix calculations establish the global properties. First, for every $h\in H$ and $w\in W$, the matrix $hw$ commutes with $x$ if and only if it commutes with $y$. If $g=u h w v$ with $u,v\in U$, both $u$ and $v$ centralize $x,y$, so this equivalence holds with $g$ in place of $hw$. Hence $C_G(x)=C_G(y)$.

Second, writing $x^g=g^{-1}xg$, the representative products $x x^{hw}$ have orders dividing one of

Indeed, torus conjugation changes the nonzero parameter of $z(t)$, leaving only seven parameters and sixteen Weyl representatives to check. For $g=u h w v$, the product $x x^g$ is conjugate to $x x^{hw}$, because $u,v$ centralize $x$. Finally, simultaneous conjugation carries any pair in $x^G$ to a pair whose first member is $x$. No product of two class members can therefore have order three. The displayed numbers are power bounds sufficient for this conclusion; an exact classification of all product orders is unnecessary.

Why the ambient group is simple

The structural argument follows the usual rank-two geometry of the Ree construction. Put $B=UH$ and $P=\langle B,r\rangle$. Bruhat decomposition shows that $P$ is maximal, and its action on the associated projective lines shows that its normal core is trivial. The subgroup $Z$ is abelian, is normalized by $P$, and normally generates $G$. Root commutator identities also give $G'=G$.

These facts imply simplicity by the following short argument. Let $K\trianglelefteq G$. If $K\le P$, then $K$ lies in the trivial core of $P$. Otherwise $KP=G$ by maximality. In this case $KZ$ is normal in $G$, since it is normalized by both $K$ and $P$. Normal generation by $Z$ gives $KZ=G$, so $G/K$ is abelian. Perfectness then gives $K=G$.

The group is finite because it is a matrix group over $k$, and is nontrivial and perfect, hence nonabelian. It contains elements of orders two and three, so these are its two smallest prime divisors. Thus every hypothesis of Gorshkov's question is met. The proof establishes the required properties directly for the generated matrix group; identifying it abstractly with a named Ree group is not needed.

Sources and credit

The question is Gorshkov's, in The Kourovka Notebook, 21st edition, Problems 21.52–21.53. The Ree groups are due to Rimhak Ree. The matrix construction and its underlying simplicity theory are prior work: R. A. Wilson, A simple construction of the Ree groups of type ²F₄, J. Algebra 323 (2010), 1468–1481. Wilson credits K. Coolsaet's earlier Ree–Tits octagon construction, including explicit root elements and parabolic generators.

The centralizer behaviour and the restriction on products also have established structural counterparts: Aschbacher–Guralnick–Segev, Elementary abelian 2-subgroups of Sidki-type in finite groups, Lemma 10.2, and Revin–Zavarnitsine, On generations by conjugate elements in almost simple groups with socle ²F₄(q²)′, Lemma 8. Exchanging vertices with the same neighbours outside the pair is a standard graph argument. Nilradical's contribution is the explicit assembly into a counterexample to Problem 21.53 and its complete formal proof.

The Lean formalization and verification record credit TauCeti for reused Tits-system and Bruhat proofs, and Lean and mathlib for the supporting foundations. The result is by Nilradical v1.0.0.