I'm reading Grünbaum and Shephard's Tilings and Patterns at the moment, and am kind of lost in the brevity of their statement and proof of Statement 10.1.1 (page 524 for anyone who has the book). Paraphrasing:
If a monohedral tiling $\mathcal{T}$ is a $k$-composition of itself in a unique way then $\mathcal{T}$ must be non-periodic, i.e. does not have translational symmetry. (A tiling $\mathcal{T}_1$ is a $k$-composition of $\mathcal{T}_2$ if every tile in $\mathcal{T}_1$ is a union of $k$ tiles in $\mathcal{T}_2$, and $k>1$ is the smallest integer for which this holds.)
In the brief proof they give is the statement "Suppose there were such a translational symmetry $t$. Then uniqueness of composition implies that $t$ must also be a symmetry of the $k$-composed tiling." They then go on to use this to derive a contradiction.
But I don't understand what uniqueness of composition has to do with it, if $t$ is a translational symmetry of $\mathcal{T}$ then isn't it automatically also a symmetry of the composed tiling?