Every group is a maximal subgroup of a naturally occurring free idempotent generated semigroup |
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Authors: | Victoria Gould Dandan Yang |
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Institution: | 1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK
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Abstract: | The study of the free idempotent generated semigroup IG(E) over a biordered set E has recently received a deal of attention. Let G be a group, let \(n\in\mathbb{N}\) with n≥3 and let E be the biordered set of idempotents of the wreath product \(G\wr \mathcal{T}_{n}\) . We show, in a transparent way, that for e∈E lying in the minimal ideal of \(G\wr\mathcal{T}_{n}\) , the maximal subgroup of e in IG(E) is isomorphic to G. It is known that \(G\wr\mathcal{T}_{n}\) is the endomorphism monoid End F n (G) of the rank n free G-act F n (G). Our work is therefore analogous to that of Brittenham, Margolis and Meakin for rank 1 idempotents in full linear monoids. As a corollary we obtain the result of Gray and Ru?kuc that any group can occur as a maximal subgroup of some free idempotent generated semigroup. Unlike their proof, ours involves a natural biordered set and very little machinery. |
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