Nonnormal Quotients

TL;DR

The paper introduces a generalized quotient operation in group theory that extends the standard quotient process to arbitrary subgroups, analyzing its properties and relationship to traditional quotients.

Contribution

It defines and characterizes a new quotient operation for arbitrary subgroups, broadening the scope of group quotient concepts.

Findings

The generalized quotient is well-defined for nonnormal subgroups.

This operation relates to existing group theory concepts.

It provides a framework for understanding quotients beyond normal subgroups.

Abstract

We present a natural extension of the process of taking a group quotient to arbitrary subgroups. We first review basic concepts from group theory. This will allow us to see the relationship between our new, more general quotient operation and the standard group quotient. In particular, we will find that a naive attempt to perform the quotient process with a nonnormal subgroup actually leads to a well-defined operation which can be easily characterized with existing terminology.