On band modules and $$\tau $$-tilting finiteness

Abstract In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties band modules and their endomorphisms in module category finite dimensional algebra. As an application describe torsion classes containing modules. Furthermore, show that special biserial algebra is if only no brick. We also recover criterion for finiteness Brauer graph algebras terms graph.