On extensions of completely simple semigroups by groups

TL;DR

This paper investigates the structure of extensions of completely simple semigroups by groups, providing examples and conditions for embeddability into wreath and semidirect products, revealing limitations and possibilities in such algebraic constructions.

Contribution

It introduces a specific extension that cannot be embedded into a wreath product and establishes conditions under which extensions are embeddable into wreath or semidirect products.

Findings

An extension of a completely simple semigroup by a group may not embed into a wreath product.

Central extensions are always embeddable into a wreath product.

Any extension can be embedded into a semidirect product with a semigroup having maximal subgroups as direct powers.

Abstract

An example of an extension of a completely simple semigroup U by a group H is given which cannot be embedded into the wreath product of U by H. On the other hand, every central extension of U by H is shown to be embeddable in the wreath product of U by H, and any extension of U by H is proved to be embeddable in a semidirect product of a completely simple semigroup V by H where the maximal subgroups of V are direct powers of those of U.