Centres of GL_n and SL_n, and discrete normal subgroups

1. Centre of $GL_n$. Let $Z$ commute with all invertible matrices. Then for $i\ne j$ it commutes with $I+E_{ij}$, hence with $E_{ij}$. Now $ZE_{ij}$ has $Ze_i$ as its $j$-th column and zeros elsewhere, while $E_{ij}Z$ has the $j$-th row of $Z$ as its $i$-th row and zeros elsewhere. Comparing entries: the $(k,j)$ entry for $k\ne i$ gives $z_{ki}=0$, the $(i,l)$ entry for $l\ne j$ gives $z_{jl}=0$, and the $(i,j)$ entry gives $z_{ii}=z_{jj}$. As $i\ne j$ are arbitrary, $Z$ is diagonal with equal diagonal entries. Thus $Z$ is a scalar matrix: $Z(GL_n)=\{\lambda I:\lambda\in\mathbb{R}^\times\}$ (conversely scalar matrices are central).

Centre of $SL_n$. The same argument applies since $I+E_{ij}\in SL_n$ for $i\ne j$. So the centre consists of the scalar matrices of determinant $1$: $\lambda^n=1$ with $\lambda\in\mathbb{R}$. If $n$ is odd, $\lambda=1$ and $Z(SL_n)=\{I\}$. If $n$ is even, $\lambda=\pm1$ and $Z(SL_n)=\{\pm I\}$.

2. Discrete normal subgroups. Let $G$ be connected, $N\trianglelefteq G$ discrete (every point of $N$ is isolated in $N$, with the subspace topology). Fix $m\in N$ and define $c:G\to N$, $c(g)=gmg^{-1}$. It is well defined since $N$ is normal and continuous (into $G$, with values in $N$). Its image is a connected subset of $G$ contained in the discrete space $N$ (in the subspace topology, a connected subset of a discrete space is a point). So $c$ is constant, equal to $c(e)=m$: $gmg^{-1}=m$ for all $g$. Hence $m$ is central.

Application. In $SL_2(\mathbb{R})$ (connected) the subgroup $\{\pm I\}$ is discrete, normal and central, as predicted. Connectedness is needed: in the non-connected group $O(2)$ the cyclic group generated by $R_{2\pi/3}$ is finite (so discrete) and normal (conjugating by a reflection gives $R_{-2\pi/3}$ and conjugating by a rotation changes nothing), but it is not central, since a reflection does not commute with $R_{2\pi/3}$.

Remark. This is the first step in showing that a covering group's kernel is central: for a connected covering $\tilde G\to G$ of topological groups, the kernel is discrete normal, hence central, hence abelian.