The Grothendieck group of the stable category of symmetric special biserial algebras

Cartan matrices of symmetric special biserial algebras are described, and the order of the Grothendieck group for the stable category of these algebras is computed. In particular, a criterion for the existence of a free summand in these groups is obtained. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 5–12.