# Isometry group of Riemannian symmetric spaces and Lie groups

How do I prove the statement that for a Riemannian symmetric space $$M$$, the isometry group $$Iso(M)$$ is a Lie group? What can we say about the dimension of $$Iso(M)$$? And is its action transitive on $$M$$? And what can we say about the isometry group of a Lie group endowed with a left/right (not bi) invariant Riemannian metric (as we know if the metric is bi invariant, then the Lie group is a symmetric space)?

P.S I not only look for answers but also for resource and literature to probe such related questions. (Note: In the definition of Riemannian symmetric space that I come across, the space is not assumed to be simply connected)

• $Iso(M)$ is a Lie group is proved by Myers and Steenrod. – user10354138 Jun 7 at 6:58
• @user10354138 The poster asked about symmetric spaces, on which the isometry group does always act transitively. – Danu Jun 7 at 9:02
• Try Helgasson's book and see if you can handle it. – Moishe Kohan Jun 7 at 10:32
• Thanks a lot. What about isometry groups of lie groups with left invariant (not bi invariant) metric? – Rama Seshan Chandrasekaran Jun 13 at 3:43