I've just started to learn isometries and groups, and I'm currently learning isometries of bounded (finite) figure. I'm confused that
For a bounded figure $F \in \Bbb R^2$ and G is a set of all isometries of F. Then G is a group under composition of functions?
Say your figure $F$ is a square, centered at the origin. There are 8 linear maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ which are isometries of the square. As examples, you have reflection over the $x$-axis -- call that one $S_x$ -- and you have rotation counterclockwise by 90 degrees about the origin -- call that one $R_{90}$.
Both of these are linear maps and therefore functions, so you can write a composition of two functions $R_{90} \circ R_{90}$. This composition happens to be rotation counterclockwise by 180 degrees, which we might call $R_{180}$. Another composition $S_x \circ S_x \circ S_x$ is the same as $S_x$. No matter how many compositions you do, the result is one of 8 total functions.
A group is a set and an operation on the set. In this case the set consists of the 8 functions I mentioned (among which are $R_{90}$, $S_x$, $R_{180}$, $I$, and four others). The operation is composition of functions.
A different figure might have a different set of isometries. In any case, composing the isometries gives you another isometry, so composition of functions makes the set of isometries a group.