hw6 - ⊆ S 4 and let K = σ ∈ S 4 | σ(4 = 4(a Show that...

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Math 113 Homework 6, due 3/8/2012 at the beginning of section Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. Fraleigh section 13, exercises 4, 5, 8, 10, 19, 49. 2. Fraleigh section 13, exercises 32, 44. 3. Fraleigh section 14, exercises 31, 34, 37. (Recall that an automorphism of G is an isomorphism from G to itself. For each a G there is an automorphism i a defined by i a ( g ) = aga - 1 ; an automorphism of G is called inner if it is i a for some a G .) 4. Let H = { id, (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3)
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Unformatted text preview: } ⊆ S 4 , and let K = { σ ∈ S 4 | σ (4) = 4 } . (a) Show that H is a subgroup of S 4 . Is H is a normal subgroup? What other group is H isomorphic to? Same questions for K . (b) Show that every coset of H contains exactly one element of K . Also show that every element of S 4 can be written uniquely as the product of an element of H and an element of K . (c) What can you say about the quotient group S 4 /H ? Is S 4 isomorphic to the direct product H × K ? 5. How challenging did you ±nd this assignment? How long did it take?...
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