Given a finite Abelian group: $G=\mathbb{Z}\_{n\_1} \times \mathbb{Z}\_{n\_2} \times \mathbb{Z}\_{n\_3}$, where ${n\_1},{n\_2},{n\_3}$ are arbitrary positive integers. ${n\_1},{n\_2},{n\_3}$ may have or may not have common divisors.


**Question**: What is **the smallest number of $N$** such that the following Lie group contains $G=\mathbb{Z}\_{n\_1} \times \mathbb{Z}\_{n\_2} \times \mathbb{Z}\_{n\_3}$ as a finite discrete subgroup? i.e. $N=$?


(1) SU($N$)


(2) SO($N$)


(3) SU($N$) $\times$ SU($N$)


(4) SO($N$) $\times$ SO($N$)


eg. for $G=\mathbb{Z}\_{2} \times \mathbb{Z}\_{2} \times \mathbb{Z}\_{2}$, I suppose the answer for **the smallest number of $N$** is:


(1) SU($4$)


(2) SO($4$)


(3) SU($3$) $\times$ SU($3$)


(4) SO($3$) $\times$ SO($3$)


What will be the cases of $N$ for a generic $G=\mathbb{Z}\_{n\_1} \times \mathbb{Z}\_{n\_2} \times \mathbb{Z}\_{n\_3}$, then $N=$?


Thank you.



The compact Lie group contains a finite subgroup $\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$
As soon as a compact connected Lie group has rank at least 3, i.e. it contains a 3-dimensional torus, it will contain your group $G$ (for all values of the parameters). So for example $SU(4), SO(6), SU(3)\times SU(3),SO(4)\times SO(4)$ do the job. Of course you must see whether these values are optimal...
Where does the question come from?


