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

TL;DR

This paper provides a concise and natural proof that every group G can be realized as the maximal subgroup of a free idempotent generated semigroup, using the biordered set of idempotents of End F_n(G).

Contribution

It offers a shorter, more natural proof of Gray and Ruskuc's result, utilizing a naturally occurring biordered set from End F_n(G).

Findings

Every group G is a maximal subgroup of some free idempotent generated semigroup.

The proof uses the biordered set of idempotents of End F_n(G).

Applicable for finite groups G with free G-act of rank at least 3.

Abstract

Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of some free idempotent generated semigroup IG(E) on a biordered set of idempotents E, thus resolving a long standing open question. Given the group G, they make a careful choice for E and use a certain amount of well developed machinery. Our aim here is to present a short and direct proof of the same result, moreover by using a naturally occuring biordered set. More specifically, for any free G-act F_n(G) of finite rank at least 3, we have that G is a maximal subgroup of IG(E) where E is the biordered set of idempotents of End F_n(G). Note that if G is finite then so is End F_n(G).