mt1 - H ⊆ G be a subgroup of order 24 Suppose that there...

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C Math 250A PROFESSOR KENNETH A. RIBET First Midterm Examination September 30, 2004 12:40–2:00 PM Name: Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Explain your answers in full English sentences as is customary and appropriate. Your paper is your ambassador when it is graded. Problem: Your score: Total points 1 6 points 2 6 points 3 6 points 4 6 points 5 6 points Total: 30 points 1
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1. If n is a positive integer, find an m 1 so that the alternating group A m contains a subgroup isomorphic to the symmetric group S n . 2
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2. Prove that every group of order 312 = 2 3 · 3 · 13 has a proper non-trivial normal subgroup. 3
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3. Let G be a group of order 120, and let
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Unformatted text preview: H ⊆ G be a subgroup of order 24. Suppose that there is at least one left coset of H in G (other than H itself) that is also a right coset of H in G . Prove that H is a normal subgroup of G . 4 4. Let G be a group and let H be a subgroup of G such that the index ( G : H ) is finite. Prove that there is a normal subgroup H of G such that H ⊆ H and such that ( G : H ) is finite. Show further that there is an n ≥ 1 so that g n ∈ H for all g ∈ G . 5 5. Let G be a finite group, and let H be a normal subgroup of G . Let P be a p-Sylow subgroup of H , and let K be the normalizer of P in G . Establish the equality G = HK . 6...
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mt1 - H ⊆ G be a subgroup of order 24 Suppose that there...

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