Siberian Federal University

For a root system Φ, the set A = {Ar | r ∈ Φ} of additive subgroups Ar over commutative ring K is called a carpet of type Φ if commuting two root elements xr (t), t ∈ Ar and xs(u), u ∈ As, gives a result where each factor lies in the subgroup Φ(A) generated by the root elements xr (t), t ∈ Ar , r ∈ Φ. The subgroup Φ(A) is called a carpet subgroup. It defines a new set of additive subgroups A = {Ar | r ∈ Φ}, the name of the closure of the carpet A, which is set by equation Ar = {t ∈ K | xr (t) ∈ Φ(A)}. Ya. Nuzhin wrote down the following question in the Kourovka notebook. Is the closure A of a carpet A a carpet too? (question 19.61). The article provides a partial answer to this question. It is proved that the closure of a carpet of type Φ over commutative ring of odd characteristic p is a carpet if 3 does not divide p when Φ of type G2