On Maximal Subgroups of Free Idempotent Generated Semigroups

TL;DR

This paper demonstrates that all groups, including finite and finitely presented ones, can be realized as maximal subgroups within various classes of free idempotent generated semigroups, expanding understanding of their subgroup structure.

Contribution

It establishes that every group can be embedded as a maximal subgroup in free idempotent generated semigroups, including finite and regular cases, revealing their rich subgroup structure.

Findings

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

Every finitely presented group is a maximal subgroup of a finite semigroup's free idempotent generated semigroup.

Every finite group is a maximal subgroup of a free regular idempotent generated semigroup from a finite regular semigroup.

Abstract

We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup.