1)Let  $G$ be  a  countable  discrete group. Can $G$ be embbeded in a  locally connected Lie group?

2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metric. Can $G$ be embbeded isometrically in a locally connected Lie group with its left invariant  metric?

**Remark**: We emphasis on word  **Locally** connected because of the example $G\subset \mathbb{R}\setminus\{0 \}$ with $G=\{\pm e^n\mid n\in \mathbb{Z} \}$ with multiplication.