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 Ruškuc 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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Notes
It is more usual to identify elements of E with those of \(\overline{E}\), but it helps the clarity of our later arguments to make this distinction.
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The authors are grateful to a careful referee for some helpful comments.
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Communicated by László Márki.
Dedicated with gratitude and affection to the memory of John Howie.
Research supported by EPSRC grant no. EP/I032312/1. The authors would like to thank Robert Gray and Nik Ruškuc for some useful discussions.
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Gould, V., Yang, D. Every group is a maximal subgroup of a naturally occurring free idempotent generated semigroup. Semigroup Forum 89, 125–134 (2014). https://doi.org/10.1007/s00233-013-9549-9
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DOI: https://doi.org/10.1007/s00233-013-9549-9