On a Presentation of the Automorphism Group of a Partially Commutative Metabelian Group

Automorphisms of a partially commutative metabelian group $S_\Gamma$ defined by a finite simple graph $\Gamma$ with $r$ vertices are considered. The monoid $\mathcal P$ of matrices of order $r$ is equipped with a congruence $\approx$. It is proved that the group of automorphisms acting identically on the quotient group by the commutator subgroup $S_\Gamma/[S_\Gamma, S_\Gamma]$ is isomorphic to the quotient monoid $\mathcal P/\approx$.