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hw5 - (3 points 5 Let p be a prime let H = Z p Z ⊕ Z p Z...

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Math 5125 Friday, September 16 Fifth Homework Due 9:05 a.m., Friday September 23 1. Let G be a group and let H char K char G . Prove that H char G . (char = characteristic subgroup) (2 points) 2. Let p be a prime, let H G be finite groups, and let P be a Sylow p -subgroup of G . Prove that H P is a Sylow p -subgroup of H . (Show that p | H : H P | ; ungraded HW from Aug 22 maybe helpful.) (3 points) 3. Determine the number of nonisomorphic abelian groups of order 7200. (2 points) 4. List the elementary divisors and match them with the corresponding invariant factors for all abelian groups of order 216.
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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 non-identity 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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