Free idempotent generated semigroups and endomorphism monoids of free G-acts

Abstract The study of the free idempotent generated semigroup IG ( E ) over a biordered set E began with the seminal work of Nambooripad in the 1970s and has seen a recent revival with a number of new approaches, both geometric and combinatorial. Here we study IG ( E ) in the case E is the biordered set of a wreath product G ≀ T n , where G is a group and T n is the full transformation monoid on n elements. This wreath product is isomorphic to the endomorphism monoid of the free G -act End F n ( G ) on n generators, and this provides us with a convenient approach. We say that the rank of an element of End F n ( G ) is the minimal number of (free) generators in its image. Let e = e 2 ∈ End F n ( G ) . For rather straightforward reasons it is known that if rank e = n − 1 (respectively, n ), then the maximal subgroup of IG ( E ) containing e is free (respectively, trivial). We show that if rank e = r where 1 ≤ r ≤ n − 2 , then the maximal subgroup of IG ( E ) containing e is isomorphic to that in End F n ( G ) and hence to G ≀ S r , where S r is the symmetric group on r elements. We have previously shown this result in the case r = 1 ; however, for higher rank, a more sophisticated approach is needed. Our current proof subsumes the case r = 1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some IG ( E ) . On the other hand, varying r again and taking G to be trivial, we obtain an alternative proof of the recent result of Gray and Ruskuc for the biordered set of idempotents of T n .