Show that the isometries $\phi: M\rightarrow M$ of a surface $M$ form a group

Question

Show that the isometries $\phi: M\rightarrow M$ of a surface $M$ form a group

Wikipedia states:
The set of bijective isometries from a metric space to itself forms a group with respect to function composition, called the isometry group.

I'm not sure how to show that.

Answer 1

The isometry group of a metric space is the set of all bijective(thus have inverses) isometries from the metric space onto itself, with the function composition(associativity) as group operation. Its identity element is the identity function.

So we can see that the group laws are all satisfied.