An algorithm to recognize echelon subgroups of a free group

Abstract

We provide an algorithm that, given a finite set of generators for a subgroup [Formula: see text] of a finitely generated free group [Formula: see text], determines whether [Formula: see text] is echelon or not and, in case of affirmative answer, also computes a basis with respect to which [Formula: see text] is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.