Every group is a maximal subgroup of a naturally occurring free idempotents generated semigroup

Victoria Gould, Dandan Yang

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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∈N with n≥3 and let E be the biordered set of idempotents of the wreath product G≀Tn. We show, in a transparent way, that for e∈E lying in the minimal ideal of G≀Tn, the maximal subgroup of e in IG(E) is isomorphic to G.
It is known that G≀Tn 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.
Original languageEnglish
Pages (from-to)125-134
Number of pages10
JournalSemigroup Forum
Issue number1
Early online date27 Nov 2013
Publication statusPublished - Aug 2014


  • G-act
  • Idempotent
  • Biordered set

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