Unformatted text preview: (3 points) 5. Let p be a prime, let H = Z / p Z ⊕ Z / p Z , and deﬁne k : H → H by k ( a , b ) = ( a + b , b ) . (a) Show that k ∈ Aut ( H ) and that k has order p . (b) Let K = h k i , let φ : K → Aut ( H ) denote the natural inclusion, and let G = H o φ K . Show that G is a nonabelian group of order p 3 , and if p is odd then every nonidentity element of G has order p . Indicate at what point you use the hypothesis that p 6 = 2. (3 points) (5 problems, 13 points altogether)...
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 Fall '07
 PALinnell
 Math, Algebra, Group Theory, Normal subgroup, Prime number, char G. Prove

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