Embedding $(\mathbb{R},+)$ into $(\text{Sym}(\omega),\circ)$

Question

Is there an injective group homomorphism from $(\mathbb{R},+)$ into $(\text{Sym}(\omega),\circ)$, where $\text{Sym}$ denotes the set of all bijections $f:\omega\to\omega$? If not, is there a group homomorphism from $(\mathbb{R},+)$ into $(\text{Sym}(\omega),\circ)$ with countable kernel?

Answer 1

As an abstract group, $\mathbf{R}$ is isomorphic to $\mathbf{Q}^\omega$.

Since $\mathrm{Sym}(\omega)^\omega$ embeds into $\mathrm{Sym}(\omega^2)\simeq\mathrm{Sym}(\omega)$, we deduce that any countable (unrestricted) direct product of countable groups embeds into $\mathrm{Sym}(\omega)$.

Actually any abelian group of cardinal $\le c$ embeds into $\mathrm{Sym}(\omega)$.