2012/09/06 by Victoria Gould, Gould, Victoria, Dandan Yang +1
Computer Science · Mathematics · #semigroups and automata theory #Geometric and Algebraic Topology #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1209.1242
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 Fn(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 Fn(G). Note that if G is finite then so is End Fn(G).