Are permutations of columns and rows on symmetric matrices continuous?

Question

Consider the group $G$ of actions to permute simultaneously rows and columns on symmetric and real matrices $A_{m \times m}$.

I wonder if $G$ is acting continuously on $A_{m \times m}$.

Thanks in advance.

Answer 1

If I understand the quesiton well, the answer is yes: Since the group $G$ is discrete, continuity of the action reduces to the fact that for any fixed element $g\in G$, the action of $g$ is continuous as a map $A_{m\times m}\to A_{m\times m}$. If you identify $A_{m\times m}$ with $\mathbb R^{m(m+1)/2}$, say via the entries $a_{ij}$ for $i\leq j$, then the action of $g$ becomes a permutation of cooridnates, so it is continuous.