Maximal subgroups of free idempotent generated semigroups over the full transformation monoid

TL;DR

This paper proves that the maximal subgroup of the free idempotent generated semigroup over the full transformation monoid, containing a specific idempotent with image size r, is isomorphic to the symmetric group S_r.

Contribution

It establishes a precise isomorphism between maximal subgroups and symmetric groups for certain idempotents in free idempotent generated semigroups over T_n.

Findings

Maximal subgroups are isomorphic to symmetric groups S_r.

The result applies to idempotents with image size r < n-1.

Provides a classification of maximal subgroups in this context.

Abstract

Let T_n be the full transformation semigroup of all mappings from the set {1,...,n} to itself under composition. Let E = E(T_n) denote the set of idempotents of T_n and let e be an arbitrary idempotent satisfying |im(e)|=r < n-1. We prove that the maximal subgroup of the free idempotent generated semigroup over E containing e is isomorphic to the symmetric group S_r.