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200a-f11-hw2

200a-f11-hw2 - H E K and K E G it does not follow in...

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Math 200a Fall 2011 Homework 2 Reading assignment: Section 6.3 (if haven’t ﬁnished), 4.1-4.3. Exercises: All exercise numbers refer to Dummit and Foote, 3rd edition. I like to list extra exercises from the text, but only the exercises marked with a star are to be handed in for grading. I include these extra exercises because they seem interesting or useful but I cannot include them without having too many assigned exercises. So please at least look over the unstarred exercises. I also frequently assign some exercises not from the text; usually these are all starred and thus to be handed in. Section 6.3 #1, 4, 7*, 11* Section 4.1 #1, 2*, 3*, 7*, 8, 9* Section 4.2 #7*, 8*, 10, 14 Exercises not from the text: (all to be handed in): The property “is a normal subgroup of” is not transitive in general; that is, if one has
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Unformatted text preview: H E K and K E G , it does not follow in general that H E G . Recall that an automorphism of a group G is an isomorphism from G to itself. We say that a subgroup H ≤ G is characteristic in G , and write H char G , if for all automorphisms φ of G , φ ( H ) = H . 1. * (a). Show that if H char G , then H E G . (b). Let H < K < G , where H char K and K E G . Then H E G . (c). Show that if K is a cyclic subgroup of G and K E G , then every subgroup H of K satisﬁes H E G . 2. * Let f,g : R → R be the functions deﬁned by the formulas f ( x ) =-x and g ( x ) = x +1. Let G = h f,g i be the subgroup of the group S R of all permutations of the set R which is generated by f and g . Prove carefully that G ∼ = h a,b | b 2 = e,ba = a-1 b i . (This group is called the inﬁnite dihedral group , A ∞ .) 1...
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