Isometries of Riemannian manifold

Question

Let $M$ be Riemannian manifold. Where I can find the proof of the following fact:

the group of isometries of $M$ is finite dimensional Lie group.

Additional question:

what is known about the dimension of this Lie group, does it depends from the choice of metric?

Answer 1

This is called "the Myers-Steenrod theorem" and you can find it right at the beginning of chapter II (page 39) in Transformation Groups in Differential Geometry by Shoshichi Kobayashi. See also the references therein.

Answer 2

See Kobayashi Nomizu Foundations of differential geometry Vol I theorem 3.4 p.239.