  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).

Rings and Algebras