The category of centralizer lattices of groups

TL;DR

This paper introduces the concept of centralizer-respecting homomorphisms in group theory, establishing a functor to centralizer lattices and exploring their categorical properties.

Contribution

It formalizes centralizer-respecting homomorphisms, constructs a functor to centralizer lattices, and investigates their categorical structure and properties.

Findings

Defined centralizer-respecting homomorphisms.

Established a functor to centralizer lattices.

Proved the category of these homomorphisms has many interesting maps.

Abstract

We formalize the concept of a centralizer-respecting homomorphism, surjective homomorphisms which are equivariant with respect to taking the centralizer of a subgroup. There is a functor from the category of centralizer-respecting homomorphisms to the category of centralizer lattices. Finally, we conclude with some theorems about centralizer-respecting homomorphisms that show that the category of centralizer-respecting homomorphisms has many interesting maps.