Notation in "proof sketch" of the Banach Tarski paradox on wikipedia

Question

I'm trying to understand the proof sketch here. In step 3 of the proof sketch we have $A_{1} = S(a)M \cup M \cup B$. My understanding is that $S(a)$ and $M$ are both sets. I have failed to understand what is meant by "$S(a)M$". Can anyone shed light upon this?

Answer 1

$S(a)$ is a certain set of maps on the unit sphere, $M$ is a subset of the unit sphere, thus it makes sense to define $S(a)M := \{s(x) | s\in S(a), x\in M\}$ i.e. as the image of every point in $M$ under every map in $S(a)$.