Mathematics & Physics 2015, 8(1), 104–108 УДК 512.54 Generation of the Chevalley Group of Type G2 over the Ring of Integers by Three Involutions Two of which Commute Ivan A.Timofeenko∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.12.2014, received in revised form 21.12.2014, accepted 24.01.2015 It is proved that G2(Z) is generated by three involutions. <...> Keywords: ring of integers, generating involutions, Chevalley group Introduction The main result of this article is Theorem 1. <...> The Chevalley group G2(Z) over the ring of integers Z is generated by three involutions and two of these involutions commute. <...> Theorem 1 answers the question formulated by Ya. <...> N.Nuzhin [1, question 15.67] for the group G2(Z) : What adjoint Chevalley groups over the ring of integers are generated by three involutions, two of which commute? <...> We just know that groups SLn(Z), n 14 are generated by three involutions, two of which commute [2]. <...> Groups PSLn(Z) are generated by three involutions, two of which commute when n 5 [3]. <...> Note also that adjoint Chevalley group B2(Z) is not generated by three involutions, two of which commute. <...> Notation and preliminary results Let Φ be a reduced indecomposable root system. <...> Let us denote adjoint Chevalley group over a field K by Φ(K). <...> Let us denote special linear group by SL2(K) and subgroup generated by the set M by M. <...> There is a homomorphism from SL2(K) onto subgroup Xr,X−r of Φ(K) such that 1 t 0 1 1 0 t 1 ∗ivan.timofeenko@gmail.com Siberian Federal University. <...> Ivan A.Timofeenko Generation of the Chevalley Group of Type G2 over the Ring. <...> With conjugations the diagonal elements act on root elements as follows: hr(t)xs(u)hr(t)−1 = xs(tArs u), (1) where Ars = 2(r, s)/(r, r) and (x, y) is the scalar product of vectors x, y. <...> Let H be a diagonal subgroup of a group Φ(K) generated by elements hr(t), r ∈ Φ, t <...>