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Problem 4. Let V denotes the vector space Rntm with basis e1 Consider the following bi-linear form on V + - - - Let O(m,n n) denote the set of all matrices A in GLm+n which preserve the form B, i.e., for any vectors u. e E V and SO(m.n) be the set of matrices A in O(m,n which also satisfy the condition det(A) = 1. a) Show that O(m. n) and SO(m, n) are subgroups of Glatai b) Find a condition which is satisfied by all matrices in O(m,n); c) Explain why the Lie algebras of O(m. n) and SO(m,n) are the same; d) Describe the Lie algebra so(m,n) of O(m,n); Problem 2. Let G denotes the group GL,(R) for n > 2. Let H be a normal subgroup of G. a) Find all possibilities for H assuming it is discrete; b) Find all possibilities for H assuming it is closed subgroup, i.e., if lim h, E H and h, - g for some g E G, then g € H; c) Give an example of non closed normal subgroup; d) Describe all normal subgroups of GL"(R). e) Find all values of m and n such that the Lie algebra so(m, n) is simple; f) Give (infinitely many) examples of pairs (m,n) such that the group SO(m,1 n) is not connected. Hint: For parts b) and d) you need to find conditions like AAT = Id and A + AT = 0. Part f) is harder than the rest.

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Matrix Groups Problems
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