## 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≀Tn, where G is a group and Tn is the full transformation monoid on n elements. This wreath product is isomorphic to the endomorphism monoid of the free G -act View the MathML source on n generators, and this provides us with a convenient approach.

We say that the rank of an element of View the MathML source is the minimal number of (free) generators in its image. Let View the MathML source. For rather straightforward reasons it is known that if View the MathML source (respectively, n ), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if View the MathML source where 1≤r≤n−2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in View the MathML source and hence to G≀Sr, where Sr 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 Ruškuc for the biordered set of idempotents of Tn.

We say that the rank of an element of View the MathML source is the minimal number of (free) generators in its image. Let View the MathML source. For rather straightforward reasons it is known that if View the MathML source (respectively, n ), then the maximal subgroup of IG(E) containing ε is free (respectively, trivial). We show that if View the MathML source where 1≤r≤n−2, then the maximal subgroup of IG(E) containing ε is isomorphic to that in View the MathML source and hence to G≀Sr, where Sr 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 Ruškuc for the biordered set of idempotents of Tn.

Original language | English |
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Pages (from-to) | 133-176 |

Number of pages | 44 |

Journal | Journal of Algebra |

Volume | 429 |

Early online date | 18 Feb 2015 |

DOIs | |

Publication status | Published - 1 May 2015 |

## Keywords

- Semigroup
- Maximal subgroup
- G-act
- Idempotent
- Biordered set
- Wreath product