Presentations for Subgroups of Monoids

The main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation. It is a modification of the well known Reidemeister–Schreier theorem for groups. Some consequences of this result are explored. It is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented. An inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid. An example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited.