Every group is a maximal subgroup of the free idempotent generated semigroup over a band

TL;DR

This paper constructs specific bands of idempotents such that their free idempotent generated semigroups contain maximal subgroups isomorphic to any given group, including finite groups, addressing open questions in the field.

Contribution

It introduces a method to construct bands of idempotents whose free idempotent generated semigroups have prescribed maximal subgroups, including all groups, both finite and infinite.

Findings

Constructed bands with prescribed maximal subgroups

For finitely presented groups, the bands are finite

Addresses open questions in the theory of idempotent generated semigroups

Abstract

Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is finite. This answers several questions from recent papers in the area.