1. The problem statement, all variables and given/known data Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). 2. Relevant equations 3. The attempt at a solution So I know the properties that define a group are - Set with binary operation (in this case matrix multiplication) - Axiom of Closure (operation produces another element of the set) - Axiom of Invertibility (There's an inverse (multiplicative and additive)) - Axiom of Identity (There's an identity element (multiplicative and additive)) - Axiom of Associativity (There's an associative property (additive and multiplicative)) I would imagine we need to show that the elements of GL(n,R), O(n,R) and SO(n,R) satisfy all these properties. My question comes in because I'm not able to find lists of any of the elements of these groups, just their descriptive properties. I.e., the Orthogonal group O(n,R) is the group of distance preserving transformations of euclidean n-space and can be represented by (n x n) matrices whose inverse equals their transpose. So is it possible to show all these properties are satisfied from simply working with the various properties of the groups or would I need to perform calculations on elements of the set to say show its closed under matrix mult., or show what element is the identity/inverse element, or demonstrate the associativity? Any help on how to proceed would be greatly appreciated.