We know that there is an injective homomorphism from symmetric group over $n$ symbols, $\mathrm{S}_n$ to the group of $n \times n$ invertible real matrices, $\mathrm{GL}_n(\mathbb{R})$, via the permutation matrix.
I wanted to know if this is also true for symmetric group $\mathrm{Sym}(X)$ over infinite set $X$ to the general linear group $\mathrm{GL}(V)$ over some arbitrary (infinite dimensional) vector space $V$. That is, does there exists an injective homomorphism between these two groups and what might it look like?
Elaborating on Arturo’s comment:
Given a set $X$, consider a vector space $V$ with basis $\{e_x\}_{x \in X}$. For each $\sigma \in \text{Sym}(X)$, define a linear map $P_\sigma \colon V \to V$ by $$ \forall x \in X, \quad P_\sigma(e_x) = e_{\sigma(x)}. $$ Thus, $P_\sigma \in \text{GL}(V)$ for every $\sigma \in \text{Sym}(X)$, and the map $$ P_{(\_)} \colon \text{Sym}(X) \to \text{GL}(V) $$ is an injective group homomorphism.
Note that if $X$ is finite, then this is the ‘same’ as working with permutation matrices.