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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 Spring '08
 OGUS
 Math, Group Theory, Normal subgroup, Cyclic group, automorphism, Fraleigh section

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