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Dimension of SO (n) and its generators - Mathematics Stack Exchange The generators of SO(n) S O (n) are pure imaginary antisymmetric n × n n × n matrices How can this fact be used to show that the dimension of SO(n) S O (n) is n(n−1) 2 n (n − 1) 2? I know that an antisymmetric matrix has n(n−1) 2 n (n − 1) 2 degrees of freedom, but I can't take this idea any further in the demonstration of the proof Thoughts?
Fundamental group of the special orthogonal group SO(n) Question: What is the fundamental group of the special orthogonal group SO(n) S O (n), n> 2 n> 2? Clarification: The answer usually given is: Z2 Z 2 But I would like to see a proof of that and an isomorphism π1(SO(n),En) → Z2 π 1 (S O (n), E n) → Z 2 that is as explicit as possible I require a neat criterion to check, if a path in SO(n) S O (n) is null-homotopic or not Idea 1: Maybe
Are $SO(n)\\times Z_2$ and $O(n)$ isomorphic as topological groups? I am doing Exercise 4-16 in Armstrong's Basic Topology The question is : are SO(n) × Z2 and O(n) isomorphic as topological groups? (I have proved the homeomorphic part) The problem is, I can prove that the map I constructed is merely a homeomorphism but not an isomorphism, but I cannot prove that there exists no isomorphism I am aware of this question, where an answer says to consider the
In a family with two children, what are the chances, if one of the . . . For example, suppose there is a social science study on 2 child families with at least 1 daughter-- in this situation, about 1 3 of the families will be daughter-daughter, 1 3 will be daughter-son, and 1 3 will be son-daughter You have to consider the full probability space of two trials (d-d,d-s,s-d,s-s) and eliminate the s-s possibility
Lie Algebra of U(N) and SO(N) - Mathematics Stack Exchange U(N) and SO(N) are quite important groups in physics I thought I would find this with an easy google search Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?
Boy Born on a Tuesday - is it just a language trick? The only way to get the 13 27 answer is to make the unjustified unreasonable assumption that Dave is boy-centric Tuesday-centric: if he has two sons born on Tue and Sun he will mention Tue; if he has a son daughter both born on Tue he will mention the son, etc
Universal covering group and fundamental group of $SO(n)$ If H H is a topological group which is both path-connected and locally path-connected (i e a connected Lie group such as SO(n) S O (n)), then any path-connected cover of H H inherits a unique group structure making the covering map a group homomorphism In fact for any such cover p: G → H p: G → H,we have ker(p) ≅π1(H) p∗(π1(G)) k e r (p) ≅ π 1 (H) p ∗ (π 1 (G)) This
Prove that the manifold $SO(n)$ is connected The question really is that simple: Prove that the manifold SO(n) ⊂ GL(n,R) is connected it is very easy to see that the elements of SO(n) are in one-to-one correspondence with the set of orthonormal basis of Rn (the set of rows of the matrix of an element of SO(n) is such a basis) My idea was to show that given any orthonormal basis (ai)n1 in Rn there's a continuous deformation from (ai