Abstract:
In 2002, the second author of this paper wrote down the following problem in the Kourovka notebook (question 15.67). A) What adjoint Chevalley groups (of normal type) over the ring of integers are generated by three involutions, two of which commute? This problem solved only for the Chevalley groups of type $A_n$ (the case $PSL_{n+1}$), $E_n$, and $G_2$. Of course, problem A) can be consider for other one-generated rings, and not only for the adjoint Chevalley groups. The analogue of problem A) is solved for the groups $PSL_{n}(\mathbb{Z}+i\mathbb{Z})$ and $SL_{n}(\mathbb{Z}+i\mathbb{Z})$ over the ring of the Gaussian integers $\mathbb{Z}+i\mathbb{Z}$, and for some small dimensions $n\leq 6$ the answer is negative. In this article we prove that the Chevalley group $G_2(\mathbb{Z}+i\mathbb{Z})$ over the ring of the Gaussian integers is generated by three involutions, two of which commute. As a consequence, the minimal number of generating involutions, whose product is equal to $1$, coincides with $5$.
Key words:Chevalley group, the ring of Gaussian integers, generating triples of involutions.
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