We introduce a concept of convergence of order $\alpha$, with $0<\alpha\leq1$, with respect to a summability matrix method $A$ for sequences (which generalizes the notion of statistical convergence of order $\alpha$), taking values in ($\ell$)-groups. Some main properties and differences with the classical $A$-convergence are investigated. A Cauchy-type criterion and a closedness result for the space of convergent sequences according our notion is proved.