Let $G$ be a connected Lie group, denote $d(G)$ as the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\mathbb{R}^q$ is isomorphic to a closed subgroup of $G$.

My question: do we always have $c(G) \leq d(G)$?

Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\mathbb{R}^q$ is isomorphic to a closed subgroup of $G$.

is there a way to compute $d(G)$ and $c(G)$?

if you have any related references, please share them whit me.

Let $G$ be a connceted Lie group, note $d(G)$ as the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\mathbb{R}^q$ is isomorphic to a closed subgroup of $G$.

My question: are we always have $c(G) \leq d(G)$?

Let $G$ be a connected Lie group, denote $d(G)$ as the maximal integer $p$ such that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\mathbb{R}^q$ is isomorphic to a closed subgroup of $G$.

My question: do we always have $c(G) \leq d(G)$?