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Algebra2009jan - ALGEBRA PHD PRELIMINARY EXAM 9 JANUARY...

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ALGEBRA PHD PRELIMINARY EXAM, 9 JANUARY 2009 1. (10 points) Let G be a group of order 132 = 2 2 · 3 · 11. Prove that G is not simple. 2. (10 points) Let H and K be normal subgroups of a group G , and assume that G = HK . Prove that there is an isomorphism G /( H K ) = G / H × G / K . (Formal manipulations with isomorphism theorems will not be enough; you’ll need to explicitly define a map.) 3. Let A be a complex square matrix of size n . (a) (3 points) Define what it means for A to be Hermitian. (b) (7 points) If X AX * has real entries for every X C n , prove that A is Hermitian. 4. (10 points) Let F be a field and V a vector space over F , not necessarily of finite dimen- sion. Let S and T be subsets of V such that S is linearly independent and T spans V . Prove that V has a basis B with S B S T . 5. A linear operator T : V -→ V , with V a finite-dimensional vector space, is called nilpo- tent if some power of T is zero. (a) (5 points) Prove that T is nilpotent if and only if its characteristic polynomial is p ( t ) = t k for some k .
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