A Note on Embedding Nonabelian Finite Flavor Groups in Continuous Groups

TL;DR

The paper discusses how nonabelian finite flavor groups embedded in SO(3) can have double covers in SU(2) that are not subgroups, providing explicit examples relevant to particle physics model building.

Contribution

It clarifies the relationship between finite flavor groups and their double covers in SU(2), with explicit examples illustrating this non-inclusion phenomenon.

Findings

Examples of $D_n$ and $Q_{2n}$ groups demonstrating the double cover relationship.

Explicit cases of $T$ and $T'$ groups showing the non-inclusion.

Implications for particle physics model construction.

Abstract

A nonabelian finite flavor group $G \subset SO(3)$ can have double covering $G^{'} \subset SU(2)$ such that $G \not\subset G^{'}$. This situation is not contradictory, but quite natural, and we give explicit examples such as $G=D_n, G^{'}=Q_{2n}$ and $G=T, G^{'}=T^{'}$. This observation can be crucial in particle theory model building.