Abstract

Let $T_n$
be the full transformation semigroup of all mappings from the set
$\{1, \dots, n\}$
to itself under composition. Let $E = E(T_n)$
denote the set of idempotents of $T_n$
and let $e \in E$
be an arbitrary idempotent satisfying $|\mathrm{im}(e)|=r \leq n-2$.
We prove that the maximal subgroup of the free idempotent generated semigroup over $E$
containing $e$ is isomorphic to the symmetric group $S_r$.