200a-f11-hw1 - exercise 12 that R Z is isomorphic to the...

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Math 200a Fall 2011 Homework 1 Due Friday 9/30/2011 by 5pm in homework box Reading assignment: Review Chapters 1-3 of Dummit and Foote, and read Section 6.3. I suggest you go carefully over the proofs of the isomorphism theorems in Section 3.3. 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 1.1 #25*, 31 Section 1.3 # 15*, 16, 17, 19* Section 1.6 #6, 7, 18, 25, 26 Section 2.1 #6, 7 Section 2.4 # 14, 18*(c,d only),19 Section 3.1 #5, 12, 14*(c,d only) (suggestion for (d): prove using the homomorphism of
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Unformatted text preview: exercise 12 that R / Z is isomorphic to the circle group S 1 , i.e. the subgroup of C * consisting of elements of norm 1, then look at the torsion subgroups of both sides), 15, 36*, 42 Section 3.2 #9*, 20*, 21 Section 3.3 #3, 7* (suggestion: use the first isomorphism theorem). Exercises not from the text: (all to be handed in): 1*. Suppose that G = H ∪ K , where H and K are subgroups of G . Show that either G = H or G = K (i.e., no group is the union of two proper subgroups.) 1 2*. Suppose that G is a finite group and that G = H 1 ∪ H 2 ∪ H 3 , where each H i is a proper subgroup of G . Show that | G : H i | = 2 for all i . Also, find an example where this actually happens. (Hint: first show by counting that at least one of the subgroups, say H 1 , has index 2. Then prove that this forces H 1 H i = G and | H i : H 1 ∩ H i ] = 2, for i = 2 , 3, and do some more counting.) 2...
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