Set invariant under group action

Question

I am reading a paper with the following description:

1. $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$

We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. So according to the article

1. $T$ should be $(U,V)\cdot X$
2. $UXV^T\in \mathbf{R}^{n\times n}$; therefore, $\mathbf{R}^{n\times n}$ is invariant under the group action.

Why does the paper says "This action leaves the set $O(n)$ invariant"? $UXV^T\notin O(n)$.