# hw6 - d be a divisor of n(a Show that H = R,R d,R 2 d is a...

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Math 113 Homework # 6, due 10/21/9 at 2:10 PM 1. Fraleigh section 14 exercises 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 .) 2. Compute the following quotient groups in terms of the classiﬁcation of ﬁnitely generated abelian groups: (a) Z Z / h (6 , 9) i (b) Z Z / h (4 , 2) , (0 , 2) i 3. Fraleigh section 15 exercises 14, 19, 23. 4. Let n be a positive integer and let
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Unformatted text preview: d be a divisor of n . (a) Show that H = { R ,R d ,R 2 d ,... } is a normal subgroup of D n . (b) Describe the cosets of H in D n . (c) Show that D n /H ’ D d . (d) Show that the commutator subgroup of D 2 n is { R ,R 2 ,R 4 ,... } . (e) Show that the abelianization D ab 2 n of D 2 n is isomorphic to Z 2 × Z 2 . 5. How challenging did you ﬁnd this assignment? How long did it take?...
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## This note was uploaded on 02/16/2010 for the course MATH 113 taught by Professor Ogus during the Spring '08 term at Berkeley.

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