Indecomposable (1,3)-Groups and a matrix problem

Almost completely decomposable groups with a critical typeset of type (1, 3) and a p-primary regulator quotient are studied. It is shown that there are, depending on the exponent of the regulator quotient pk, either no indecomposables if k ⩽ 2; only six near isomorphism types of indecomposables if k = 3; and indecomposables of arbitrary large rank if k ⩾ 4.