# Math Help - centre

1. ## centre

why is it we say that the centre of SU(N) is isomorphic to Z (mod n) and center of U(N) is isomorphic to U(1) when
both their centres are {aI : lal=1}

are they different or the same

2. Originally Posted by alexandrabel90
why is it we say that the centre of SU(N) is isomorphic to Z (mod n) and center of U(N) is isomorphic to U(1) when
both their centres are {aI : lal=1}

are they different or the same
Well it's almost always true (as a rule of thumb--not a theorem) that given a (complex, although more general fields of characteristic zero work) matrix group $G$ one has that $\mathcal{Z}(G)=G\cap \left\{zI:z\in\mathbb{C}\right\}$ and this is true for $\text{SU}(n)$ so that $\mathcal{Z}\left(\text{SU}(n)\right)=\left\{zI:z\i n\mathbb{C}\right\}\cap\text{SU}(n)$. That said, we know that since $zI\in\text{SU}(n)$ it's true that $1=\det(zI)=z^n$ so that $z_n\in\mu_n$ (this is a common notation for the $n^{\text{th}}$ roots of unity) and moreover you can check that $zI\in\mathcal{Z}\left(\text{SU}\left(n\right)\righ t)$ for every $z\in\mu_n$ so that $\mathcal{Z}\left(\text{SU}(n)\right)\cong\mu_n$ but it's basic group theory that $\mu_n\cong\mathbb{Z}_n$. Does that help?